578
Recent Advances in Orthogonal Polynomials, Special Functions, and Their Applications
11th International Symposium August 29–September 2, 2011 Universidad Carlos III de Madrid Leganés, Spain
J. Arvesú G. López Lagomasino Editors
American Mathematical Society
Recent Advances in Orthogonal Polynomials, Special Functions, and Their Applications
11th International Symposium August 29–September 2, 2011 Universidad Carlos III de Madrid Leganés, Spain
J. Arvesú G. López Lagomasino Editors
578
Recent Advances in Orthogonal Polynomials, Special Functions, and Their Applications
11th International Symposium August 29–September 2, 2011 Universidad Carlos III de Madrid Leganés, Spain
J. Arvesú G. López Lagomasino Editors
American Mathematical Society Providence, Rhode Island
EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor George Andrews Abel Klein Martin J. Strauss
2000 Mathematics Subject Classification. Primary 30E05, 30E10, 30E15, 33-XX, 42C05, 42C10, 41A20, 41A21, 41A25, 41A30.
Library of Congress Cataloging-in-Publication Data International Symposium on Orthogonal Polynomials, Special Functions and Applications (11th : 2011 : Universidad Carlos III de Madrid) Recent advances in orthogonal polynomials, special functions, and their applications : 11th International Symposium on Orthogonal Polynomials, Special Functions, and Their Applications, August 29–September 2, 2011, Universidad Carlos III de Madrid, Leganes, Spain / J. Arves´u, G. L´opez Lagomasino, editors. p. cm. — (Contemporary Mathematics ; v. 578) Includes bibliographical references. ISBN 978-0-8218-6896-6 (alk. paper) 1. Functions of complex variables–Congresses. I. Arves´u, Jorge, 1968– II. L´opez Lago- masino, Guillermo, 1948– III. Title.
QA331.7.I596 2011 515 .9–dc23 2012017031
Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Math- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) c 2012 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Copyright of individual articles may revert to the public domain 28 years after publication. Contact the AMS for copyright status of individual articles. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 171615141312
Dedicated to Francisco (Paco) Marcell´an on the occasion of his 60th birthday
Contents
Preface ix Life and work (so far) of Paco Marcell´an Manuel Alfaro and Walter Van Assche 1
Asymptotics of Lp-norms of Hermite polynomials and R´enyi entropy of Rydberg oscillator states A. I. Aptekarev, J. S. Dehesa, P. Sanchez-Moreno,´ and D. N. Tulyakov 19 The next-order term for optimal Riesz and logarithmic energy asymptotics on the sphere J. S. Brauchart, D. P. Hardin, and E. B. Saff 31 Spectral transformations of hermitian linear functionals M. J. Cantero, L. Moral, and L. Velazquez´ 63 Numerical study of higher order analogues of the Tracy–Widom distribution T. Claeys and S. Olver 83 Comb functions A. Eremenko and P. Yuditskii 99 Orthogonality relations for bivariate Bernstein-Szeg˝omeasures J. S. Geronimo, P. Iliev, and G. Knese 119 Quantum walks and CMV matrices F. Alberto Grunbaum¨ 133 Discrete beta ensembles based on Gauss type quadratures D. S. Lubinsky 143 Heine, Hilbert, Pad´e, Riemann, and Stieltjes: John Nuttall’s work 25 years later A. Mart´ınez-Finkelshtein, E. A. Rakhmanov, and S. P. Suetin 165 Orthogonal polynomials and S-curves E. A. Rakhmanov 195 Fast decreasing and orthogonal polynomials V. Totik 241
vii
Preface
This volume contains a selection of papers presented at the 11th International Symposium on Orthogonal Polynomials, Special Functions and their Applications (OPSFA’11), held from August 29 to September 3, 2011, at Universidad Carlos III de Madrid, Legan´es, Spain. Most of them reflect the contents of the talks delivered by the plenary speakers. The conference, as well as this volume, was dedicated to celebrate the 60th birthday of Professor Francisco Marcell´an Espa˜nol who is a distinguished member of the Spanish mathematical community and has done a tremendous job in placing Spain at the head of research in the areas covered by the conference. OPSFA’11 is the eleventh edition of a series of conferences which started at Bar-le-Duc, France, in 1984. See http://matematicas.uc3m.es/index.php/ opsfa-history for a complete list of these conferences and their history. They have played a major role throughout these years in keeping the subject alive and produc- ing major results. On this occasion, for the first time the Szeg˝o Prize was awarded. This prize, instituted by the SIAM Activity Group on Orthogonal Polynomials and Special Functions (SIAG/OPSF), is intended for early career researchers with relevant contributions in the area. The recipient was Tom Claeys of the Catholic University of Louvaine and this volume contains a contribution of his. The conference had 213 participants, which is a record for the series, from over 35 countries. There were 15 plenary speakers. Two of these talks were dedicated to outlining Francisco Marcell´an’s contributions in the field and are included in the opening paper of the book. Additionally, 100 contributed talks were given and 40 posters were exhibited. The papers presented here contain new results and methods, recent develop- ments, and new trends as well as a selection of open problems which will foster interest in research in Orthogonal Polynomials, Special Functions, and their Appli- cations in the coming years from both theoretical and applied perspectives. As co-organizers of OPSFA’11 and editors of this volume it is our duty to thank those individuals and institutions whose efforts made it possible. Most of all, we acknowledge Ministerio de Ciencia e Innovaci´on of Spain (grant MTM2010- 12283-E), Proyecto Ingenio Mathematica (grant SARE-C6-0426), Universidad Car- los III de Madrid (grant 2011/00342/001), Universidad Polit´ecnica de Madrid (grant VORG-08/10), Instituto de Ciencias Matem´aticas (grant 200450E564), Real So- ciedad Matem´atica Espa˜nola, Sociedad Espa˜nola de Matem´atica Aplicada, and the Society for Industrial and Applied Mathematics (SIAM) for their financial support. Second, it is a pleasure to thank all the members of the Organizing Committee for the excellent organization of this meeting as well as the members of the Scientific Committee who helped us to make an excellent selection of invited speakers and
ix
xPREFACE chaired the plenary talks. Last but not least, we express our gratitude to the par- ticipants of the workshop who made this a memorable event, to the contributors of this volume, and to Christine Thivierge of the AMS staff for her efficient support in the production of these proceedings.
Jorge Arves´u, Universidad Carlos III de Madrid Guillermo L´opez Lagomasino, Universidad Carlos III de Madrid.
Contemporary Mathematics Volume 578, 2012 http://dx.doi.org/10.1090/conm/578/11476
Life and work (so far) of Paco Marcell´an
Manuel Alfaro and Walter Van Assche Dedicated to the 60th anniversary of Paco Marcell´an
Abstract. We give a short description of the life and work of Francisco (Paco) Marcell´an. First we present some aspects of Paco’s life related to his ini- tial years, studies, activities, hobbies, etc. Next we will make an attempt to describe his scientific contributions. This includes orthogonal polynomials on Cassinian curves (lemniscates), modifications of orthogonal polynomials, Sobolev orthogonal polynomials, recurrence relations and differential equa- tions, matrix orthogonal polynomials, semi-classical orthogonal polynomials, etc. Special attention will be paid to his pivotal role as a coordinator and public relations officer of orthogonal polynomials in Spain.
Life of Paco Marcellan´ 1. The early years It is very difficult to summarize in a few pages a life so full of activities and events as Paco’s life. But let’s start at the beginning: Francisco (Paco) Marcell´an was born September 15, 1951, in the city of Zaragoza, Spain. His full name is Francisco Jos´e Marcell´an Espa˜nol, where, as is customary in Spain, Marcell´an cor- responds to his father’s family name and Espa˜nol to his mother’s family name. This full name is only used in official documents and on forms. As in many countries, the usual way to call someone is to use a short name. A peculiarity of Paco is that he has two short names: for his relatives he is Paco Pepe (Paco for Francisco and Pepe for Jos´e) and for the rest of the world he is Paco. The first years of Paco’s life are marked by the position of his father Jos´eMar´ıa Marcell´an Alcubierre. Jos´eMar´ıa Marcell´an studied medicine at the University of Zaragoza, where he finished his studies in 1932. In the subsequent years, he worked as a doctor, but this job was interrupted by the Spanish Civil War (1936–39) in which he participated as a volunteer. At the end of the war, he decided to remain in the army and, despite of his medical training, he was stationed in the Service (or Quartermaster) Corps. At the end of the forties, he was appointed to Zaragoza
2010 Mathematics Subject Classification. Primary 33C45, 42C05; Secondary 15A24. Key words and phrases. Orthogonal polynomials, semi-classical functionals, Sobolev orthogonality. M.A. is supported in part by MICINN of Spain under Grant MTM2009-12740-C03-03 and the DGA project E-64 (Spain). W.V.A. is supported by KU Leuven Research Grant OT/08/033 and FWO Grant G.0427.09.
c 2012 American Mathematical Society 1
2 MANUEL ALFARO AND WALTER VAN ASSCHE where he occupied several positions, one of them in the Academia General Militar (Spanish Military Academy) as a teacher. In 1950, he married Alicia Espa˜nol D´ıaz in Zaragoza. They had a son (Paco) and a daughter, Maite (Mar´ıa Teresa), born in Zaragoza on the 2nd of March, 1954. At that time, the officers of the Spanish Army moved very frequently and so Paco spent his childhood in several places: Zaragoza (1951–56), Melilla (1956–57), Avila´ (1957–58), and finally in the town of Jaca, on the outskirts of the Pyrenees mountains, where his family arrived in 1958 and stayed for a long time. There, Paco’s father passed away in 1966. In Jaca, Paco studied at primary and secondary schools: primary and the first half of secondary school (1957–1965) at the Colegio de las Escuelas P´ıas (College of the Pious Schools), a catholic school, and the second half of the secondary school (1965–68) at the Instituto Nacional de Ense˜nanza Media “Domingo Miral” (National Institute of Secondary Education “Domingo Miral”), a state school.
2. At the university Once he finished his high school, someone suggested him to study at the Univer- sidad de Deusto, a catholic university located in the Basque Country and specialized in Business and Management, but Paco, fortunately, decided to study mathemat- ics. So, in October 1968, he moved to Zaragoza and he enrolled in the Faculty of Sciences at the University of Zaragoza. He was an excellent student, had very good qualifications, and was awarded several prizes from the University of Zaragoza and on local and national scale. In particular he received in 1974 the Premio Nacional de Licenciatura en Matem´aticas given each year to the best student in mathematics among all the Spanish universities. Paco has good memories of many of his teachers and he usually cites Prof. Mar´ıa Pilar Alfaro (calculus), the late Prof. Juan Sancho (algebra) and Prof. Jos´eGaray (functional analysis), but above all he was influenced very much by the lectures of the late Prof. Luis Vigil (functions of several real variables and functions of a complex variable). During his studies in mathematics, Paco lived in Pedro Cerbuna Hall (Colegio Mayor “Pedro Cerbuna”), a residence for students and professors located on the campus of the university. There several activities were organized and Paco used to participate very actively in some of them. Besides sports, which we will talk about later, he attended meetings organized by Christian groups, connected with the Catholic Parish of the University, where the social and political conditions in Spain were analyzed. At that time, Spain was under a dictatorial government that did not respect human rights and carried out a heavy police repression. The opposi- tion to the regime of general Franco was increasing among workers and intellectuals. The Spanish universities were not strange to that atmosphere and between teach- ers and students a great dissatisfaction and a political effervescence raised. Paco became more and more involved in it and the position evolved into a libertarian ideology. As a consequence he joined in 1972 the historical libertarian Trade Union Confederaci´on Nacional del Trabajo (National Labour Confederation, CNT). The CNT was often persecuted by the police and so Paco had some trouble and, in par- ticular, was held under custody for questioning at several occasions. His syndical activism continued after he finished University and during the period 1975–85 he was very much involved with these activities. He was the Secretary General of the
LIFE AND WORK (SO FAR) OF PACO MARCELLAN´ 3
CNT in Arag´on (the region of Spain whose capital is Zaragoza) from June 1977 to February 1978 and, once he moved to Madrid, he was the person in charge of the international relations of the Trade Union from December 1983 to June 1987. Nowadays he still is a member but not a militant and he collaborates with the jour- nal of opinion and reflection on social reality “Libre Pensamiento” (Free Thought), supported by the Trade Union CGT (Confederaci´on General del Trabajo). Simultaneous to this political activity, he developed social activities in suburbs of Zaragoza, giving literacy courses and general education courses for people of low economic level, mainly old aged people and women of workers. Paco is very proud of this activism that he held until he left Zaragoza in 1981.
3. Doctoral dissertation Paco began to work on his Doctoral dissertation at the end of 1973 after grad- uating in mathematics in June 1973. He received a solid mathematical formation during the five years spent at the Faculty and, as a consequence of the lectures of Prof. Vigil, he was strongly attracted by Mathematical Analysis and in particular by Complex Analysis. During the academic year 1972–73 he collaborated as a stu- dent in the department of Theory of Functions with a research grant for initiation. This allowed him to get in contact with the research group supervised by Luis Vigil. Prof. Luis Vigil y V´azquez, founder of the Spanish School on Orthogonal Polynomials, first worked at Complutense University in Madrid (1942–1959) and then at the Central University of Caracas in Venezuela (1959–1966). Then he got a position as catedr´atico (Full Professor) at the University of Zaragoza in January of 1967 and immediately started to organize a research group on his two fields of interest: Fourier Analysis and Orthogonal Polynomials. When Paco joined the department, Jos´e Luis Rubio de Francia (Fourier Analysis) and Mar´ıa Pilar and Manuel Alfaro, Jaime Vinuesa, and Enrique Atencia (Orthogonal Polynomials) were working on their respective doctoral dissertations. When Paco contacted Vigil, who was trying to move to Complutense University of Madrid, Vigil asked Prof. Jos´e Luis Rubio de Francia to tutor Paco. Jos´eLuis Rubio de Francia, an excellent young researcher specialized in Fourier Analysis, proposed to Paco a problem on convergence in measure. The result was Paco’s first paper [28] (in collaboration with Jos´e Luis) which was presented at the First Spanish-Portuguese Mathematical Conference (Madrid, 1973) and published in the corresponding Proceedings. Paco was very happy to collaborate with Rubio de Francia and was considering to ask him to be his supervisor. But then the situation changed: Rubio de Francia obtained a two year grant for Princeton University and Vigil decided to stay in Zaragoza. Because of this, Paco finally began his research under the supervision of Vigil. Some years before, Vigil had suggested in [43] to study orthogonal polynomials on real algebraic curves, in particular to develop a parametric theory like Geronimus did for orthogonal polynomials on the unit circle. So one of the topics that he pro- posed to his students was the study of orthogonal polynomials on algebraic curves in the complex plane as a generalization of the two well known models: orthogonal polynomials on the real line {z ∈ C : z =0} and orthogonal polynomials on the unit circle {z ∈ C : |z| =1}. Consequently, the interest was focused on two families of curves: harmonic algebraic curves, (A(z)) = 0, and lemniscates, |A(z)| = c, where A is a polynomial and c is a real positive constant. The Ph.D. thesis of Paco
4 MANUEL ALFARO AND WALTER VAN ASSCHE was dedicated to orthogonal polynomials on lemniscates, with the assumption that the polynomial A has only simple roots, which was a generalization of orthogonal polynomials on Bernoulli’s lemniscate, studied by Atencia. It is worthy to note that some of the usual techniques for orthogonal polyno- mials on the real line or on the unit circle do not work for orthogonal polynomials on algebraic curves. Paco used other techniques to analyze the problem than those used in previous papers about orthogonal polynomials on curves (Szeg˝o, Smirnov, Keldysh). Paco’s main tools were the multiplication operator by the polynomial A and several orthogonal decompositions of the space of polynomials of degree less than or equal to n, in order to obtain suitable bases which lead to different representations of the orthogonal polynomials. In his dissertation, recurrence and summation formulas were obtained, the density of the polynomials in L2 and some questions about Fourier series and Jacobi series were analyzed. The dissertation ends with a very interesting appendix including several open problems that were later studied by Paco and his students. The Ph.D. thesis entitled Polinomios or- togonales sobre cassinianas (Orthogonal polynomials on Cassinians) was defended in December 1976. As we have said, Paco considered the situation when the poly- nomial A has simple zeros but he later suppressed this restriction until he achieved a beautiful theory that he and some of his collaborators developed over a period of more than twelve years in about 30 papers.
Figure 1. From left to right: Rosa, Clara, Paco’s mother, Paco and Alba in 2000
During this period, one of the most important events in Paco’s life took place. In the summer of 1976 he met Rosa Fern´andez Cifuentes for the first time and she became his lifelong companion. Rosa and Paco were married in Zaragoza on September 15, 1978. They have two lovely daughters: Alba and Clara both born in Zaragoza, on November 1, 1979 and December 30, 1981, respectively. Rosa has always been the main support of Paco, helping and encouraging him at all time. Moreover, since Paco has no driver’s license, Rosa has always been his personal driver.
LIFE AND WORK (SO FAR) OF PACO MARCELLAN´ 5
4. Research career Returning to Paco’s research career, it should be noted that at that time the mathematical research in Spain began to take off and develop. Until then there had been little contact with foreign mathematicians and the papers, often written in Spanish, were published in Spanish mathematical journals and conference proceed- ings, usually without a selection process by referees. As Renato Alvarez-Nodarse,´ one of his students, claims: the research of Paco has had an evolution parallel or similar to that of the Spanish mathematical research. In fact, for many years Paco published his papers in that way. The first paper in a refereed journal, a joint work with Andr´e Ronveaux, appeared in the Canadian Mathematical Bulletin in 1989 [27]. Since then, the published work of Paco is impressive: he has published more than 200 papers in peer reviewed journals, with more than 100 co-authors. Furthermore he has edited several conference proceedings and has been co-author of several textbooks. In fact, it is a bit difficult to give a precise number of his pub- lications because when looking in the usual databases, one quickly realizes that the number of Paco’s publications increases almost daily. One fact that Paco regrets is that he never published a joint paper with his supervisor Luis Vigil. So far Paco has supervised 7 Master and 31 Ph.D. theses. Of his students 21 were Spanish, 3 Portuguese, 2 from Cuba, and 1 from Colombia, Mexico, Venezuela, Morocco, and Kosovo each. The first international meeting that Paco attended was the International Con- gress of Mathematicians held in Helsinki, Finland, in 1978. He remembers that he saw Chihara’s book there for the first time. Before that moment his main references had been the books of Szeg˝o and Freud and books of some Soviet math- ematicians such as Geronimus, Akhiezer, Smirnov, and Lebedev. Two years later, at the suggestion of Jes´us S´anchez Dehesa, Paco went to a meeting of the American Mathematical Society in Ann Arbor, Michigan, where a special session on orthog- onal polynomials was organized. There he came in touch with some USA mathe- maticians specialized in orthogonal polynomials: Askey, Nevai, Ismail, Ullman, and Geronimo, among others. Paco also gave his first presentation out of Spain: a short communication, a revision of which was published in [18]. Since then his research career began to take off with many new contacts and an international projection. In September 1981, at the VI Meeting of the Groupement des Math´ematiciens d’Expression Latine, he came in touch with Andr´e Ronveaux from Namur, Belgium. This contact was renewed a few years later in Bar-le-Duc, France, at the occasion of the first International Symposium on Orthogonal Polynomials and Applications and it was the beginning of a long and fruitful collaboration. In August 1983, at the occasion of the International Congress of Mathematicians held in Warsaw, Poland, he met Guillermo L´opez and Evguenii Rakhmanov. Paco has been Visiting Professor at many Universities and Mathematical Re- search Centers in Spain and abroad. His first long stay outside Spain was in 1987 at the Universit´e Pierre et Marie Curie (Paris VI), invited by Pascal Maroni, with whom he had come in contact in Bar-le-Duc. In Bar-le-Duc he also met Claude Brezinski, Andr´e Draux, Alphonse Magnus, and some other French and Belgian mathematicians. His main contacts are in Coimbra (Portugal), Lille, Paris VI and Paris VII (France), Namur and Leuven (Belgium), Columbus (Ohio) and Atlanta (Georgia), Krakow (Poland), and Linz (Austria).
6 MANUEL ALFARO AND WALTER VAN ASSCHE
5. Professional career During his professional career, Paco has combined teaching and research. From Zaragoza, where he taught at the Faculty of Sciences (1972–1974) and at the High Technical Engineering School (1974–1981), he moved as Full Professor to the High Technical Engineering School of the Universidad de Santiago de Compostela (1981– 82), Universidad Polit´ecnica de Madrid (1982–1991) and finally to Universidad Carlos III de Madrid (1991 until now). An anecdotal note: Paco, as his father, also lectured at the Spanish Military Academy, where the Faculty of Sciences of Zaragoza was responible for the first course of Physics for the future Spanish offi- cers. Everybody knows that he is an excellent lecturer who transmits very well the mathematical ideas and concepts, and above all, he is an enthusiastic teacher.
Figure 2. Ambition as an administrator, already in 1958–1959
His teaching and scientific activities have been complemented by an intense participation in university management. At present he is the Head of the Depart- ment of Mathematics at Universidad Carlos III de Madrid, and between 1991 and 1995 he was the first Head of the Department of Engineering. He held successively the posts of Vice-rector for Research at Universidad Carlos III de Madrid (1995– 2004), Director of the National Agency for Quality Assessment and Accreditation (ANECA) (2004–06), and Secretary General for Scientific and Technological Policy (2006–2008). Paco has also been actively involved in the SIAM Activity Group on Orthogonal Polynomials and Special Functions as programme Director (1999– 2004) and Chair (2008–present), and in the Royal Spanish Mathematical Society as Member of the Executive Committee (2000–2006).
6. Non-academic activities Paco’s life doesn’t end in the world of mathematics, but it is full of other activities. The non-academic activity in which Paco has spent most time is in sports. Curiously, Paco has never played football (soccer) although it is the most popular sport in Spain. However, he is interested in the Spanish Football League, supporting Real Zaragoza F´utbol Club and Jacetano Club de F´utbol, the most representative teams of Zaragoza and Jaca. Furthermore, despite the fact that Jaca, the city where he had lived for a long time, is close to the Pyrenees with good winter sports equipment, Paco has never practiced winter sports regularly.
LIFE AND WORK (SO FAR) OF PACO MARCELLAN´ 7
The beginning of Paco as an athlete was in the Institute of Jaca, where he played handball and basketball, participating in School Championships. Once he moved to Zaragoza he continued playing basketball, but in 1969 some students in Pedro Cerbuna Hall founded a rugby team to participate in the University Championships, and Paco entered this team where he played as a third line. He also played with the team of the Faculty of Sciences who were the champions of Arag´on two years. After some time rugby practice was not consistent with the academic work (too much time for training, matches, travel) and, consequently, Paco decided to leave rugby and replace it by running. He began to run alone, through the streets and the parks of Zaragoza, three or four days per week. In Madrid he got in contact with an athletic club in his neighborhood (Agrupaci´on Deportiva Ciudad de los Poetas), where someone suggested him to start training for running a marathon. In 1988 he participated for the first time in Madrid’s marathon with a time of 3 hours 37 minutes, quite good for a beginner. So Paco decided he should be training for the marathon in a more systematic way. Since then he has participated in 20 marathons: 15 in Madrid, four in Columbus, Ohio, jointly with Paul Nevai, and one in the North of Spain. His record is 3 hours 27 minutes in the 1991 Madrid marathon. He has also participated uninterruptedly in the 13 editions of the Intercampus races, a race which crosses the different campuses of the Universidad Carlos III. Participants in meetings of orthogonal polynomials are used to find Paco running when the sessions have finished, often together with Paul Nevai, Renato Alvarez-Nodarse´ or other people attending the conference. Amazingly, Paco still finds time for daily marathon training, running approximately 10 kilometers every day.
Figure 3. A marathon in Columbus, Ohio, with Paul Nevai in 1999
His favorite relaxing activity is to go to the movies which he enjoys every week, accompanied by Rosa, and many times in the company of friends with whom he meets on weekends to chat, after the projection, over some tapas and Spanish wine or Belgian bier, for which he is a pretty good expert. He also likes traveling, always with Rosa, and visiting exotic places. Furthermore he is a voracious reader. His main preferences are economics, politics, and essay books, together with newspa- pers, of which he usually reads two or three daily (El Pa´ıs, El Mundo, P´ublico, Le Monde Diplomatique).
8 MANUEL ALFARO AND WALTER VAN ASSCHE
Work of Paco Marcellan´ 7. Cassinian curves and lemniscates As was mentioned before, Paco’s Ph.D. thesis (1976) was titled Polinomios ortogonales sobre cassinianas (Orthogonal polynomials on Cassinians) with Luis Vigil y V´azquez (1914–2003) as the advisor. In this dissertation, Paco studied orthogonal polynomials on special algebraic curves known as Cassinian curves: Definition 7.1. A Cassinian curve is the level curve of a polynomial P {z ∈ C : |P (z)| = C>0} Sometimes the terminology lemniscate (Latin/Greek for ribbon) is used, but this is usually reserved for those Cassinian curves which are self intersecting, such as Bernoulli’s lemniscate {z ∈ C : |z2 − 1| =1}. For smaller values of the constant one gets Cassini ovals {z ∈ C : |z2 − 1| < 1}.
A Cassinian curve is also an equipotential curve [25]: if P (z)=(z −a1) ···(z −aN ), then |P (z)| = C is equivalent with N 1 log = − log C. |z − a | j=1 j Hence a Cassinian curve contains all the points for which the product of the dis- tances to N given points is constant.
Figure 4. The Cassinian curves |z2 − 1| = C with C>1(outer curve), C = 1 (Bernoulli’s lemniscate) and C<1 (Cassini ovals).
In his dissertation Paco assumed that the zeros of P are simple. Note that a Cassinian curve is the inverse polynomial image of the unit circle. Paco used this observation and his main tool was the multiplication operator A : f → Pf,where P is the polynomial defining the Cassinian curve. Paco then cleverly constructs various bases in the space Πn of polynomials with degree ≤ n,inparticularthe basis formed by the polynomials n 2 qk(z)= Pj (z)Pj(ak)/ej , j=0 where (Pn)n∈N are monic orthogonal polynomials on the Cassinian curve {z ∈ C : |P (z)| = C}, en is the norm of Pn and a1,...,aN are the zeros of the polynomial P .
LIFE AND WORK (SO FAR) OF PACO MARCELLAN´ 9
The polynomials (qk)1≤k≤N form a basis of the linear space of polynomials orthog- onal to P Πn−N . By orthogonalizing this basis one can then find simple recurrence relations expressing P (z)Pn−N (z) − Pn(z)intermsoftheN polynomials in the ba- sis, hence giving a finite order recurrence relation. Other topics considered in the thesis are summation formulas and Fourier series involving orthogonal polynomials on Cassinians. The thesis was in Spanish and most of the papers resulting from his dissertation (e.g., [3]) were in Spanish and in proceedings of conferences or local journals. As a result, most of Paco’s work on orthogonal polynomials on Cassinians or lemniscates is not as known as it should be, even though the Spanish community was quite fa- miliar with it. In fact Paco was one of the first to deal with orthogonal polynomials on inverse images of polynomials mappings, a topic which received a lot of interest later, since such sets are quite useful when one wants to study orthogonal polyno- mials on Julia sets (relevant in the iteration of polynomials) or on several intervals. Furthermore, Hilbert proved the following theorem (see, e.g., [37]) Theorem 7.2 (Hilbert). The boundary Γ of any simply connected bounded domain G can be approximated arbitrarily well by a Cassinian curve (lemniscate). Hence, when one wants to investigate orthogonal polynomials on the boundary Γ, one may benefit a lot from results for orthogonal polynomials on Cassinian curves. This approach has been used by Peherstorfer [38]andTotik[41], but for polynomial inverse images of the interval [−1, 1] rather than the unit circle. For more details about Paco’s work on orthogonal polynomials on algebraic curves we warmly recommend the contribution of Leandro Moral [35] in the Selected Works [1].
8. Bar-le-Duc: the start of OPSFA The first international meeting on Polynˆomes Orthogonaux et Applications was held in Bar-le-Duc, France in October of 1984 at the occasion of the 150th anniver- sary of Laguerre’s birthday. The meeting was organized by C. Brezinski, A. Draux, Al. Magnus, P. Maroni and A. Ronveaux and it was the first of a series of interna- tional meetings on Orthogonal Polynomials and their Applications. Later Special Functions were also added as a topic. Most of the experts working on orthogonal polynomials attended this conference and Paco Marcell´an was among them. Two papers were published in the proceedings: a paper on recurrence formulas for or- thogonal polynomials on Bernoulli’s lemniscate [23], with his first Ph.D. student Leandro Moral, and one on a Christoffel formula for orthogonal polynomials on a Jordan curve [12], with Paloma Garc´ıa-L´azaro, a master student who later also prepared a Ph.D. under Paco’s supervision. The meeting in Bar-le-Duc was very successful and it was decided to plan the next meeting in Spain in 1986. This second meeting on Orthogonal Polynomials and their Applications was organized by M. Alfaro, J.S. Dehesa, F.J. Marcell´an, J.L. Rubio de Francia and J. Vinuesa, who were all former Ph.D. students of Luis Vigil. The meeting was in Segovia in September of 1986 and for many people it was nice to return to Segovia in 2011 during the excursion of the 11th meet- ing on Orthogonal Polynomials, Special Functions and Applications (OPSFA-11). The proceedings were again published in the Lecture Notes in Mathematics [2]. Paco had a contribution with one of his Ph.D. students [10], this time with Alicia Cachafeiro on jump modifications of orthogonal polynomials.
10 MANUEL ALFARO AND WALTER VAN ASSCHE
Figure 5. Paco at the Bar-le-Duc meeting in France, 1984
9. Perturbations of orthogonal polynomials From the beginning, Paco was interested a lot in various modifications of or- thogonal polynomials. Suppose a family of orthogonal polynomials (Pn)n∈N for a known measure μ is given. The measure μ may be on the real line, on the unit circle or on a set of the complex plane. We are then interested to construct the orthogonal polynomials (Qn)n∈N for a modification μi of the measure μ. Possible modifications of the measure μ are • N adding mass points: μ1 = μ + j=1 cj δxj ,whereδx is the Dirac measure which is concentrated at the point x; • multiplication by a polynomial: dμ2(x)=P (x)dμ(x), where P is a poly- nomial of degree N. This is now known as the Christoffel transform, since E.B. Christoffel gave a formula [40, Thm. 2.5 on p. 29] express- ing P (x)Qn(x) in terms of a determinant containing the polynomials Pn+N (x),...,Pn(x) and their values at the zeros of the polynomial P ; • dividing by a polynomial: dμ3(x)=1/Q(x) dμ(x), where Q is a polyno- mial of degree M. This is sometimes known as the Uvarov transform, since V.B. Uvarov gave a formula [16, Thm. 2.7.3 on p. 39] similar to Christoffel’s formula, but now also involving the Stieltjes transform P (x) n dμ(x) z − x evaluated at the zeros of Q; • a combination of adding a mass point at a and dividing by Q(x)=x−a,so −1 that dμ4(x)=cδa +(x−a) dμ(x). This is called a Geronimus transform in [44], referring to work of Geronimus in 1940. Observe that first applying a Geronimus transform and then a Christoffel transform with P (x)=x−a, retrieves the original measure. Conversely, first applying the Christoffel transform with P (x)=x − a and then the Geronimus transform gives the original measure to which a mass point at a is added.
LIFE AND WORK (SO FAR) OF PACO MARCELLAN´ 11
Note that the modification by means of adding mass points is sometimes also called an Uvarov transform. In fact Uvarov considered both modifications in [42], but adding mass points was done much earlier already, but it is difficult to trace the first person to write down the formulas for the modified polynomials. Paco worked on all these modifications and usually in a general setting where one does not need a measure on a set in the complex plane but orthogonality is defined in terms of a linear functional acting on polynomials. See e.g., [12]for the Christoffel transform, [21] for the Uvarov transform (in fact, a combination of Christoffel and Uvarov, which gives a modification by a rational function), and [10, 21] for jump modifications. This list is certainly not exhaustive and Andrei Mart´ınez-Finkelshtein [33] describes Paco’s work on these canonical transforma- tions (and the related Darboux transformation) in somewhat more detail in the Selected Works [1].
10. Sobolev orthogonal polynomials Moving on to the next international conference, we arrive in Columbus, Ohio in May–June of 1989 where P. Nevai had succeeded to organize a two week NATO Advanced Study Institute on Orthogonal Polynomials and Their Applications.One particular talk was very important in the further career of Paco, namely the talk Orthogonality in a Sobolev space. Originally Arieh Iserles was supposed to give the talk, but in his absence the talk was given by Lisa Lorentzen. The talk corresponds to the paper [15] of Iserles et al. and deals with orthogonal polynomials with respect to a Sobolev inner product Sn(x)Sm(x) dμ1(x)+λ Sn(x)Sm(x) dμ2(x)=δm,n, where (μ1,μ2) is a pair of positive measures on the real line and λ>0 a constant.
Figure 6. Paco in action during the NATO ASI in Columbus, Ohio, 1989
Sobolev orthogonal polynomials were already investigated a couple of times before (by Althammer in 1962, Brenner in 1972, Cohen in 1975, Gr¨obner in 1967, and Sch¨afke in 1972, to name a few) but with little impact. The new idea introduced
12 MANUEL ALFARO AND WALTER VAN ASSCHE by Iserles et al. was to choose the pair of measures (μ1,μ2) cleverly so that the Sobolev orthogonal polynomials (Sn)n∈N can be expressed in a somewhat easier way in terms of the orthogonal polynomials (pn)n∈N for μ1 and (qn)n∈N for μ2.
Definition 10.1. The pair (μ1,μ2)iscoherentifandonlyifthereexistnon- zero constants (Cn)n∈N such that the orthogonal polynomials for μ1 and μ2 are related by − qn(x)=Cn+1pn+1(x) Cnpn(x),n=1, 2,....
For a coherent pair (μ1,μ2) one can write the Sobolev orthogonal polynomials as n −1 Sn(x)= αk(λ)pk(x) − βn(λ)pn(x) k=1 where the αn(λ) obey a three-term recurrence relation and βn(λ) is a simple expres- sion of αn and αn−1. This new idea was picked up quickly by Paco and by a team of mathematicians from Delft. Paco saw a whole set of open problems for Sobolev or- thogonal polynomials and he encouraged many people in Spain to work on Sobolev orthogonal polynomials. The paper [4] with Alfaro, Rezola and Ronveaux is one of his most cited papers. One challenging problem was to classify all coherent pairs. Many cases were investigated where μ1 or μ2 were chosen as a classical measure (Jacobi, Laguerre, Hermite) and the corresponding Sobolev orthogonal polynomi- als were investigated by means of differential equations, recurrence relations and algebraic properties. The complete classification of coherent pairs was finally given by Henk Meijer [34]. This was for classical weights where one uses the differen- tial operator, but Paco observed that one could introduce inner products involving other operators, such as the difference operator [8]ortheq-difference operator [7] and together with Ivan Area and Eduardo Godoy, he investigated coherent pairs for such Sobolev-type inner products. A very important property of orthogonal polynomials on the real line is the three-term recurrence relation, which gives rise to the Jacobi operator. For or- thogonal polynomials on the unit circle one has the Szeg˝o recurrences, which give rise to a Hessenberg matrix (the GGT representation [39, §4.1 on p 251], named after Geronimus, Gragg and Teplyaev), which is unitary outside the Szeg˝o class, or a special pentadiagonal matrix (the CMV representation [39, §4.2 on p. 262], named after Cantero, Moral and Vel´azquez). These matrix representations of the multiplication operator M : f → Mf for which Mf(z)=zf(z)areawaytoinves- tigate orthogonal polynomials using spectral theory. Unfortunately, no such matrix representation has been found for Sobolev orthogonal polynomials, and this is one of the main drawbacks of the theory of Sobolev orthogonal polynomials. There are some results, with important contributions of Paco, that tell us that a matrix representation for some multiplication operator will only exist for a restricted class of Sobolev orthogonal polynomials, namely those for which the derivatives in the inner product are only evaluated at a finite number of points (i.e., the measure μ2 is a finite discrete measure).
Theorem 10.2 (Evans et al. [11]). Suppose there exists a polynomial h of degree ≥ 1 satisfying hp, q = p, hq , for polynomials p and q, where the inner
LIFE AND WORK (SO FAR) OF PACO MARCELLAN´ 13 product is of the form
N (k) (k) p, q = p (x)q (x) dμk(x). R k=0
Then the measures μk, 1 ≤ k ≤ N are necessarily of the form
mk
(10.1) μk = αk,jδxk,j . j=1 The corresponding Sobolev orthogonal polynomials satisfy a recurrence relation of the form
n +m (10.2) h(x)Sn(x)= bn,kSk(x) k=n−m where m isthedegreeofh. Conversely, if the measures μk, 1 ≤ k ≤ N in the Sobolev inner product are of the form (10.1), then there exists a unique (up to a constant multiple) polynomial h of minimal degree m ≥ N +1 such that hp, q = p, hq and the recurrence relation (10.2) holds.
A simple but illustrative case was worked out in detail in [31], namely the inner product 1 f,g = f(x)g(x) dμ(x)+λf (c)g(c), −1 where c ∈ R and μ is in the Nevai class M(0, 1), i.e., the coefficients in the three- term recurrence relation
0 0 0 xpn(x)=an+1pn+1(x)+bnpn(x)+anpn−1(x), for the orthonormal polynomials for the measure μ have the asymptotic behavior 0 → 0 → →∞ an 1/2andbn 0asn . The relative asymptotic behavior of the ratio Sn(x)/pn(x) was obtained for x ∈ C \ [−1, 1] and the recurrence relation for the Sobolev orthogonal polynomials, as described by Theorem 10.2, is
2 (x − c) Sn(x)=an+2Sn+2(x)+bn+1Sn+1(x)+cnSn(x)+bnSn−1(x)+anSn−2(x), for which the recurrence coefficients have the asymptotic behavior 1 1+2c2 lim an = , lim bn = −c, lim cn = . n→∞ 4 n→∞ n→∞ 2 Essentially this means that the matrix representation corresponds to a penta- diagonal matrix which is a compact perturbation of (J − cI)2,whereJ is the Jacobi matrix for the orthogonal polynomials corresponding to the measure μ.A more general case, containing higher order derivatives as in (10.1), was considered in [17]. For a more detailed account of Paco’s papers in the theory of Sobolev orthogonal polynomials one may check the contribution of Juan Jos´e Moreno-Balc´azar [36]in the Selected Works [1].
14 MANUEL ALFARO AND WALTER VAN ASSCHE
11. Other topics Paco has done quite a lot of research on many problems in the theory of or- thogonal polynomials, and it is impossible to give details of all contributions. We only mention a few of the topics, such as matrix orthogonal polynomials on the unit circle [26] and on the real line [29], algebraic and functional analytic approach of orthogonal polynomials, in particular semiclassical orthogonal polynomials [19] [30] (see also the contribution of Renato Alvarez-Nodarse´ and Jos´eCarlosPetron- ilho [6] in the Selected Works [1]), electrostatic interpretations of zeros [22](see also [33] in the Selected Works [1]), orthogonal polynomials as special functions (differential equations [27], estimations [24]).
Figure 7. Paco lecturing on Laguerre-Sobolev orthogonal poly- nomials in Leuven, 1996
One result in the theory of orthogonal polynomials on the unit circle deserves more attention. Classical orthogonal polynomials on the real line are well known and there is a very nice table (Askey’s table) in which all these classical orthogonal polynomials (of hypergeometric or basic hypergeometric type) are collected, to- gether with their properties and their relations with each other. Surprisingly there are not that many explicit families of orthogonal polynomials on the unit circle. Paco and Pascal Maroni considered the Hahn characterization of classical orthogo- nal polynomials: which families of orthogonal polynomials have the property that the derivatives form a sequence of orthogonal polynomials? On the real line one gets the (very) classical orthogonal polynomials of Jacobi, Laguerre and Hermite. On the unit circle the result is much more restrictive.
Theorem 11.1 (Marcell´an and Maroni [20]). Let (ϕn)n∈N be a sequence of monic orthogonal polynomials on the unit circle and let ψn(z)=ϕn+1(z)/(n +1), n ≥ 0.Then(ψn)n∈N is a sequence of monic orthogonal polynomials on the unit n circle if and only if ϕn(z)=z for all n ∈ N.
For more details of Paco’s work on orthogonal polynomials on the unit circle we refer to the contribution of Alicia Cachafeiro [9] in the Selected Works [1].
LIFE AND WORK (SO FAR) OF PACO MARCELLAN´ 15
12. Paco as an organizer Right now, Paco is the chair of the SIAM activity group on Orthogonal Polyno- mials and Special Functions. But since the beginning of his career, he was always active as an organizer of various events involving orthogonal polynomials or as an editor of books and proceedings dealing with Orthogonal Polynomials [13]. In Spain he was one of the first main organizers of the series Simposium sobre Polinomios Ortogonales y Aplicaciones (SPOA) which brought together researchers in Spain. The first six of these symposia were: ISPOA:Logro˜no, March 1983; II SPOA: Jaca, June 1984; III SPOA: Segovia, 13–15 June, 1985. This meeting served as a tryout for the international meeting in Segovia in 1986; IV SPOA: Laredo, 7–12 September, 1987; VSPOA:Vigo, 8–10 September, 1988; VI SPOA: Gij´on, September, 1989; VII SPOA: Granada, September 23–27, 1991. This meeting, organized by J.S. Dehesa and his co-workers, grew out to an international meeting with 7 plenary speakers and 128 participants; VIII SPOA: Sevilla, September 22–26, 1997. Also this meeting, organized by A. Dur´an and his co-workers, was a large international meeting with 10 plenary lectures and 158 participants.
Figure 8. Paco enjoying a Belgian beer at the opening reception of OPSFA-10 in Leuven, 2009
Paco was also often a member of the Scientific Committee of the series of international conferences on Orthogonal Polynomials and their Applications,which since 1999 became conferences on Orthogonal Polynomials, Special Functions and their Applications (OPSFA). He was • on the local organizing committee of the second meeting in Segovia, Sep- tember 22–27, 1986 [2]; • on the International Advisory Board of the NATO Advanced Study Insti- tute (Columbus, Ohio, May 22–June 3, 1999);
16 MANUEL ALFARO AND WALTER VAN ASSCHE
• a plenary speaker at the third OPSFA meeting in Erice (Sicily, Italy), May 31–June 9, 1990; • on the International Scientific Committee of the conference Orthogonality, Moment Problems and Continued Franctions in Delft, The Netherlands, October 31–November 4, 1994 (it did not get an OPSFA number but is certainly considered to be part of the OPSFA meetings); • on the Scientific Committee of OPSFA-5 in Patras, Greece, September 20–24, 1999 (but was unable to attend); • on the Scientific Committee of OPSFA-6 in Rome, June 18–22, 2001; • an invited speaker at the International Conference on Difference Equa- tions, Special Functions and Applications, which was a joint conference (ICDEA-10, OPSFA-8, SIDE-6.5) in M¨unchen, Germany, July 25–30, 2005; • on the Scientific Committee (and a plenary speaker) of OPSFA-9 in Lu- miny, France, July 2–6, 2007; • on the International Scientific Committee of OPSFA-10 in Leuven, Bel- gium, July 20–25, 2009; • the guest of honor on OPSFA-11 in Legan´es (Madrid), Spain, August 29–September 2, 2011. Paco was also very much in favor of setting up summer schools for graduate students, Ph.D. students and young postdocs. The first of these summer schools was in September of 2000 in Laredo, in the North of Spain [5] and other summer schools followed in 2001 (Inzell, Germany), 2002 (Leuven, Belgium), 2003 (Coimbra, Portugal) and 2004 in Legan´es (Spain) [32]. Meanwhile, Paco also started up a new series of International Workshops on Orthogonal Polynomials (IWOP) at Universidad Carlos III de Madrid, starting in 1992 and organized every two years.
13. Epilog Surely we have not mentioned everything and we may have forgotten (inten- tionally and unintentionally) to include some contributions or certain aspects of Paco’s involvement with orthogonal polynomials. However, we should allow him to continue doing what he likes to do and in this respect this is only a description of his life and work so far. There is still so much that he can (and will) do. Happy 60th birthday Paco and many more fruitful years to come!
References [1] M. Alfaro, R. Alvarez-Nodarse,´ M.L. Rezola (Eds.), Selected Works of Francisco J. Marcell´an Espa˜nol. http://matematicas.uc3m.es/images/pacobook/pacobook.html [2] M. Alfaro, J.S. Dehesa, F.J. Marcell´an, J.L. Rubio de Francia, J. Vinuesa (Eds.), Orthogonal Polynomials and their Applications: Proceedings Segovia 1986, Lecture Notes in Mathematics 1329, Springer-Verlag, Berlin, 1988. MR973417 (89f:00027) [3] M. Alfaro Garc´ıa, F. Marcell´an Espa˜nol, F´ormulas de sumaci´on para polinomios ortogonales sobre lemniscatas (Summation formulas for orthogonal polynomials on lemniscates), Proceed- ings of the seventh Spanish-Portuguese conference on mathematics (Sant Feliu de Gu´ıxols, 1980), Publ. Sec. Mat. Univ. Aut`onoma Barcelona 21 (1980), 137–140. MR768206 [4] M. Alfaro, F. Marcell´an,M.L.Rezola,A.Ronveaux,On orthogonal polynomials of Sobolev type: algebraic properties and zeros, SIAM J. Math. Anal. 23 No 3 (1992), 737–757. MR1158831 (93g:42015)
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[5] R. Alvarez-Nodarse,´ F. Marcell´an, W. Van Assche (Eds.), Laredo Lectures on Orthogonal Polynomials and Special Functions, Advances in the Theory of Special Functions and Or- thogonal Polynomials, Nova Science Publishers, New York, 2004. MR2085856 (2005c:33001) [6] R. Alvarez-Nodarse,´ J.C. Petronilho, Contributions to the algebraic theory of orthogonal polynomials,[1], pp. 33–53. [7] I. Area, E. Godoy, F. Marcell´an, q-Coherent pairs and q-orthogonal polynomials, Appl. Math. Comput. 128 (2002), no. 2-3, 191–216. MR1891019 (2003d:33038) [8] I. Area, E. Godoy, F. Marcell´an, Δ-coherent pairs and orthogonal polynomials of a dis- crete variable, Integral Transforms Spec. Funct. 14 (2003), no. 1, 31–57. MR1949214 (2003m:33009) [9] M.A. Cachafeiro, Contributions to the theory of orthogonal polynomials on the unit circle, [1], pp. 19–32. [10] M.A. Cachafeiro, F. Marcell´an, Orthogonal polynomials and jump modifications, Orthogonal Polynomials and their Applications, Segovia, 1986 (M. Alfaro et al., Eds.), Lecture Notes in Mathematics 1329, Springer-Verlag, Berlin, 1988, pp. 236–240. MR973430 (90b:30004) [11] W. D. Evans, L.L. Littlejohn, F. Marcell´an,C.Markett,A.Ronveaux,On recurrence rela- tions for Sobolev orthogonal polynomials, SIAM J. Math. Anal. 26, No. 2 (1995), 446–467. MR1320230 (96c:42049) [12] P. Garc´ıa-L´azaro, F. Marcell´an, Christoffel formulas for N-kernels associated to Jordan arcs, Polynˆomes Orthogonaux et Applications, Bar-le-Duc, 1984 (C. Brezinski et al., Eds.), Lec- ture Notes in Mathematics 1171, Springer-Verlag, Berlin, 1985, pp. 195–203. MR838984 (87i:33026) [13] W. Gautschi, F. Marcell´an,L.Reichel(Eds.),Orthogonal Polynomials and Quadrature,Vol- ume 5 of Numerical Analysis 2000, J. Comput. Appl. Math. 127 (2001). [14] E. Godoy, F. Marcell´an, Orthogonal polynomials and rational modifications of measures, Canad. J. Math. 45, No. 5 (1993), 930–943. MR1239908 (95a:42031) [15] A. Iserles, P.E. Koch, S.P. Nørsett, J.M. Sanz-Serna, On polynomials orthogonal with respect to certain Sobolev inner products, J. Approx. Theory 65 (1991), no. 2, 151–175. MR1104157 (92b:42029) [16] M.E.H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable,Encyclope- dia of Mathematics and its Applications 98, Cambridge University Press, Cambridge, 2005 (paperback 2009). MR2191786 (2007f:33001) [17] G. L´opez, F. Marcell´an, W. Van Assche, Relative asymptotics for polynomials orthogonal with respect to a discrete Sobolev inner product, Constr. Approx. 11 (1995), 107–137. MR1323966 (96c:42051) [18] F. Marcell´an, M. Alfaro, Recurrence relations for orthogonal polynomials on algebraic curves, Portugal. Math. 42 (1983-1984), 41–51. MR796280 (86j:42030) [19] F. Marcell´an, A. Branquinho, J. Petronilho, Classical orthogonal polynomials: a functional approach, Acta Appl. Math. 34 (1994), No. 3, 283–303. MR1273613 (95b:33024) [20] F. Marcell´an, P. Maroni, Orthogonal polynomials on the unit circle and their derivatives, Constr. Approx. 7 (1991), 341–348. MR1120408 (92j:42026) [21] F. Marcell´an, P. Maroni, Sur l’adjonction d’une masse de Dirac `auneformer´eguli`ere et semi-classique, Ann. Mat. Pura Appl. (4) 162 (1992), 1–22. MR1199643 (94e:33014) [22] F. Marcell´an,A.Mart´ınez-Finkelshtein, P. Mart´ınez-Gonz´alez, Electrostatic models for zeros of polynomials: old, new, and some open problems, J. Comput. Appl. Math. 207,No.2 (2007), 258–272. MR2345246 (2008h:33017) [23] F. Marcell´an, L. Moral, Minimal recurrence formulas for orthogonal polynomials on Bernoulli’s lemniscate,Polynˆomes Orthogonaux et Applications, Bar-le-Duc, 1984 (C. Brezinski et al., Eds.), Lecture Notes in Mathematics 1171, Springer-Verlag, Berlin, 1985, pp. 211–220. MR838986 (87g:42041) [24] F. Marcell´an, B.P. Osilenker, Estimates of polynomials orthogonal with respect to the Legendre-Sobolev inner product, Mat. Zametki 62 (1997), no. 6, 871–880 (in Russian); trans- lation in Math. Notes 62 (1997), no. 5-6, 731–738. MR1635170 (99e:42036) [25] F. Marcell´an, I. P´erez-Grasa, The moment problem on equipotential curves, Nonlinear Nu- merical Methods and Rational Approximation, Antwerp, 1987 (A. Cuyt, Ed.), Math. Appl. 43, Reidel, Dordrecht, 1988, pp. 229–238. MR1005361 (90f:30042) [26] F. Marcell´an Espa˜nol, I. Rodr´ıguez Gonz´alez, A class of matrix orthogonal polynomials on the unit circle, Linear Algebra Appl. 121 (1989), 233–241. MR1011740 (90i:30009)
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[27] A. Ronveaux, F. Marcell´an, Differential equation for classical-type orthogonal polynomials, Canad. Math. Bull. 32 (1989), 404–410. MR1019404 (90j:33014) [28] F. Marcell´an Espa˜nol, J.L. Rubio de Francia, Funcionales continuos para la convergencia en medida (Functionals continuous with respect to convergence in measure), Actas Primeras Jornadas Matem´aticas Hispano-Lusitanas (Madrid, 1973), Consejo Sup. Inv. Cient., Madrid, 1977, pp. 140–162. MR581076 (82d:28008) [29] F. Marcell´an, G. Sansigre, On a class of matrix orthogonal polynomials on the real line, Linear Algebra Appl. 181 (1993), 97–109. MR1204344 (94b:42012) [30] F. Marcell´an, F.H. Szafraniec, Operators preserving orthogonality of polynomials, Studia Math. 120 (1996), no. 3, 205–218. MR1410448 (97j:47053) [31] F. Marcell´an, W. Van Assche, Relative asymptotics for orthogonal polynomials with a Sobolev inner product, J. Approx. Theory 72 (1993), 193–209. MR1204141 (94h:42037) [32] F. Marcell´an, W. Van Assche (Eds.), Orthogonal Polynomials and Special Functions: Compu- tation and Applications, Lecture Notes in Mathematics 1883, Springer-Verlag, Berlin, 2006. MR2233607 (2007c:33012) [33] A. Mart´ınez-Finkelshtein, Some other topics from the theory of orthogonal polynomials in Paco’s work,[1], pp. 71–82. [34] H.G. Meijer, Determination of all coherent pairs, J. Approx. Theory 89 (1997), no. 3, 321– 343. MR1451509 (99c:42046) [35] L. Moral, Contributions to the theory of orthogonal polynomials over algebraic curves,[1], pp. 13–18. [36] J.J. Moreno-Balc´azar, Contributions to the theory of Sobolev orthogonal polynomials,[1], pp. 55–70. [37] B. Nagy, V. Totik, Sharpening of Hilbert’s lemniscate theorem,J.Anal.Math.96 (2005), 191–223. MR2177185 (2006g:30008) [38] F. Peherstorfer, Deformation of minimal polynomials and approximation of several inter- vals by an inverse polynomial mapping, J. Approx. Theory 111 (2001), no. 2, 180–195. MR1849545 (2002g:41009) [39] B. Simon, Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory, Amer. Math. Soc. Colloq. Publ. 54-1, Amer. Math. Soc., Providence, RI, 2005. MR2105088 (2006a:42002a) [40] G. Szeg˝o, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ. 23,Amer.Math.Soc., Providence, RI, 1939 (fourth edition 1975). [41] V. Totik, Asymptotics for Christoffel functions for general measures on the real line,J.Anal. Math. 81 (2000), 283–303. MR1785285 (2001j:42021) [42] V.B. Uvarov, The connection between systems of polynomials that are orthogonal with respect to different distribution functions Zh. Vycisl. Mat. i Mat. Fiz. 9 (1969), 1253–1262 (in Rus- sian); translated in USSR Comput. Math. Math. Phys. 9 (1969), no. 6, 25–36. MR0262764 (41:7369) [43] L. Vigil, Polinomios ortogonales sobre curvas algebraicas reales (Orthogonal polynomials on real algebraic curves), Actas de la Und´ecima Reuni´on Anual de Matem´aticos Espa˜noles, Murcia, 1970, Univ. Complutense de Madrid, Madrid, 1973, pp. 58–70. MR0354681 (50:7159) [44] A. Zhedanov, Rational spectral transformations and orthogonal polynomials, J. Comput. Appl. Math. 85 (1997), 67–86. MR1482157 (98h:42026)
Departamento de Matematicas´ and IUMA, Universidad de Zaragoza, c/ Pedro Cer- buna 12, ES 50009 Zaragoza, Spain E-mail address: [email protected] Department of Mathematics, KU Leuven, Celestijnenlaan 200B box 2400, BE 3001 Leuven, Belgium E-mail address: [email protected]
Contemporary Mathematics Volume 578, 2012 http://dx.doi.org/10.1090/conm/578/11469
Asymptotics of Lp-norms of Hermite polynomials and R´enyi entropy of Rydberg oscillator states
A. I. Aptekarev, J. S. Dehesa, P. S´anchez-Moreno, and D. N. Tulyakov
Abstract. The asymptotics of the weighted Lp-norms of Hermite polynomi- als, which describes the R´enyi entropy of order p of the associated quantum oscillator probability density, is determined for n →∞and p>0. Then, it is applied to the calculation of the R´enyi entropy of the quantum-mechanical probability density of the highly-excited (Rydberg) states of the isotropic os- cillator.
1. Introduction A long standing problem in classical analysis and approximation theory is the determination of the weighted Lp-norms 1 1 p p ≡ p 2 p (1.1) ρn p [ρn(x)] dx = ω(x)yn(x) dx ; p>0, Δ Δ where {yn(x)} denotes a sequence of real polynomials orthogonal with respect to the weight function ω(x) on the interval Δ so that
yn(x)ym(x)ω(x)dx = δm,n; m, n ∈ N, Δ and 2 (1.2) ρn(x)=ω(x)yn(x).
We call (1.2) Rakhmanov’s probability density of the polynomial yn(x) since this mathematician discovered in 1997 (see [20]) that it governs the asymptotic (n →∞) behaviour of the ratio yn+1/yn for general ω>0 a.e. (positive almost everywhere) on the finite interval Δ. Physically, ρn(x) describes the radial probability density of the ground and excited states of the physical systems whose non-relativistic wavefunctions are controlled by the polynomials yn(x)[12]. The Lp-norms (1.1) are closely related to the frequency or entropic moments [30, 24, 13] p−1 p p Wp[ρn]= [ρn(x)] = [ρn(x)] dx = ρn p, Δ
2000 Mathematics Subject Classification. Primary 11B37, 94A17; Secondary 30E15, 33C45. Key words and phrases. Asymptotic behaviour of solutions of difference equations, orthogo- nal polynomials, Hermite polynomials, R´enyi entropy, Rydberg states.
c 2012 American Mathematical Society 19
20 A. I. APTEKAREV, J. S. DEHESA, P. SANCHEZ-MORENO,´ AND D. N. TULYAKOV the R´enyi entropies [23] 1 (1.3) R [ρ ]= ln W [ρ]; p>0,p=1 , p n 1 − p p and the R´enyi spreading lengths [15] − p R p−1 Lp [ρn]=exp(Rp[ρn]) = ρn p . These quantities have been applied in numerous fields from economics and electrical engineering to chemistry, quantum physics and approximation theory, as summarized in e.g. Refs [16,6,8,9,7]. The study of the Lp norms of orthogonal polynomials is of independent interest in the theory of general orthogonal and extremal polynomials. This problem is connected with the classical research of S.N. Bernstein on the asymptotics of the Lp extremal polynomials in [5], that has recently received further development [17], [18]. On the other hand, its statement is a generalization of a widely known problem of Steklov on the estimation of the L∞ norms of polynomials orthonormal with respect to a positive weight (see [27]). Indeed, for p = 1 the norms are bounded (they are just equal to 1); however, for p = ∞ (as it has been shown by Rakhmanov [21]) they may grow to infinity. What happens with the boundedness of the Lp-norms of the Rakhmanov density of the orthogonal polynomials when 1
2. Lp-norms of Hermite polynomials: Asymptotics (n →∞)
In this section we find the main term of asymptotics Wp[ρn−1](see(1.4))when n →∞. Here we consider Hermite polynomials Hn(x) in the standard normaliza- tion − n , 1−n 1 (2.1) H (x)=(2x)n F 2 2 ; − . n 2 0 − x2
ASYMPTOTICS OF Lp-NORMS OF HERMITE POLYNOMIALS AND RENYI´ ENTROPY 21
They satisfy the recurrence relation
Hn+1(x)=2xHn(x) − 2nHn−1(x),H0(x)=1,H−1(x)=0, and have the following norm +∞ √ 2 −x2 n (2.2) hn = Hn(x) e dx = πn!2 . −∞
To reach the goal we need to have good asymptotics for Hn(x)onR. The first strong asymptotics formulae for Hermite polynomials are due to Plancherel and Rotach [19](seealso[28]). They describe polynomials Hn when n →∞in the following subdomain of R: √ ≤ ∞ a) x = √2n +1coshy, y< ; − 1 (2.3) b) x = √2n +1+n 6 t, t ∈ K ⊂ C; c) x = 2n +1cosy, ≤ y ≤ π − . Using Plancherel–Rotach formulae (and their generalization for Freud weights from [22]) it was obtained [2] for the orthonormal Hermite polynomials H˜n−1(x)that +∞ − 2 p 1−p 4 ˜ 2 x 2 ≤ (2.4) Hn−1(x)e dx = cp(2n) (1 + o(1)) , for p , −∞ 3 where 2 p Γ(p + 1 ) Γ(1 − p ) (2.5) c = 2 2 . p 3 − p π Γ(p +1) Γ( 2 2 ) Using (2.2) and the Stirling’s formula, this expression together with (1.4) produces the following asymptotics for the entropic moment of the orthogonal Hermite poly- nomial Hn−1(x): (2.6) p 1−p − − 2 p p(n 1)+1/2 pn Wp[ρn−1]=cp hn−1 (2n) (1 + o(1)) = cpπ (2n) e (1 + o(1)) . The restriction on p in (2.4) appeared because the Plancherel–Rotach formulae in (2.3) do not match each other, i.e. subdomains in (2.3) do not intersect. Particu- larly, there is a gap between zone a) and zone b) in (2.3), which plays an important role for the limit (1.4) when n →∞. In the asymptotics of (2.4) the main contri- bution in the left hand side integral gives the part of the integral described in a) of (2.3). The gap between zone a) and zone b) gives the main contribution in the integral for bigger p. The asymptotic description of Hermite polynomials in the subdomains covering all R was obtained not so long ago. In 1999 Deift et al [11](seealso[10]) have obtained the global asymptotic portrait of polynomials orthogonal with respect to exponential weights by means of the powerfull√ matrix Riemann-Hilbert method. As a collorary for Hermite polynomials Hn( 2nz), they obtained asymptotics as n →∞and z belongs to a) |z|≥1+δ; (2.7) b) 1 − δ ≤|z|≤1+δ; c) |z|≤1 − δ, for small δ>0. Evidently, there are no gaps between zones and Deift et al’s asymptotics can be used for obtaining asymptotics of (1.4) for bigger p.
22 A. I. APTEKAREV, J. S. DEHESA, P. SANCHEZ-MORENO,´ AND D. N. TULYAKOV
Recently a new approach for obtaining the global asymptotic portrait of or- thogonal polynomials has appeared [29]. Contrary to the matrix Riemann-Hilbert method which starts from the weight of orthogonality, the starting point in [29]is the recurrence relation which characterizes the orthogonal polynomials. The appli- cation of this approach to Hermite polynomials brought an asymptotic description in the following subdomains of R: 2 1 +θ a) x ∈ [2n + n 3 ; ∞); 2 1 +θ 1 +θ (2.8) b) x ∈ [2n − n 3 ;2n + n 3 ]; 2 1 +θ c) x ∈ [0; 2n − n 3 ]. ∈ 2 for θ (0; 3 ). It is worth noting that zones a) and c) in (2.8) are wider than zones a) and c) in (2.7); in its turn zone b) in (2.7) is wider that b) in (2.8). In these 1 zones we take θ<6 for Hermite polynomials. Then, it follows from Theorem 5 of [29]that: (2.9) √ n− 1 √ 1 x + x2 − 2n 2 x2 − n − x x2 − 2n in a) Hn−1(x)=√ √ exp (1 + o(1)) , 2 4 x2 − 2n 2 √ n− 1 √ 2n 2 x2 − n − √ in c) Hn 1(x)= 2 4 exp 2 n − x2 2√ 1 x2 x 2n − x2 π × cos n − arcsin 1 − − − + o(1) (1 + o(1)) , 2 2n 2 4 √ √ 2 3 2π n− 2 x 2 − 1 in b) Hn−1(x)= √ x 3 exp + o(1) Ai − z + o n 3 (1 + o(1)) , 6 2 4 2
4 2n 3 where z := 2 − x , and Ai denotes the Airy function (see [1], page 367). x 3 Using these asymptotics we can obtain the main result of this section.
Theorem 2.1. Let Hn(x) be the Hermite polynomials with the standard nor- malization ( 2.1). Then the frequency or entropic moments Wp[ρn−1], given by Eq. ( 1.4), have for n →∞the following asymptotic values ⎧ p p(n−1)+1/2 −pn ⎨⎪ cpπ (2n) e (1 + o(1)) ,p<2, 2n− 3 −2n (2.10) Wp[ρn−1]= 2(2n) 2 e (ln(n)+O(1)) ,p=2, ⎩⎪ −p p(n− 2 )− 1 −pn 2Cp 2 (2n) 3 6 e (1 + o(1)) ,p>2. where the constant cp is defined in ( 2.5) and the constant Cp is equal to √ p +∞ 3 √2π 2 −z 2 Cp = 3 Ai dz . −∞ 2 2
We note that the first asymptotic formula in the right hand side of (2.10) coincides with (2.6), but now it holds true in the maximal range of p (when p =2, then cp = ∞); let us also highlight that the main term of the asymptotics is growing. Moreover, the smaller terms contain a constant which depends on p,and when p → 0 this constant tends to infinity; however, our formula is correct for any small fixed p>0. We also note that the leading term of all three formulae in the right hand side of (2.10) match each other when p → 2.
ASYMPTOTICS OF Lp-NORMS OF HERMITE POLYNOMIALS AND RENYI´ ENTROPY 23
Proof. Doing identical transformations and some evident asymptotic esti- mates, we have from (2.9) that: (2.11) n−1 2 2 −x (2n) −n x 2 in a) H − (x)e = e exp (2n − 1)arccosh√ − x x − 2n n 1 2 2n − 1 × x2 − 2 2n 1 (1 + o(1)) ; 2 −x2 n−1 −n in c) Hn−1( x)e =(2n) e √ × − − − x2 − − 2 1 sin (2n 1) arcsin 1 2n x 2n x + o(1) − 1 × − x2 2 1 2n (1 + o(1)) ; − 2 2(n 3 ) 2 − 2 − 2 − 2π x 2n − x 2 x n 3 n √ √ in b) Hn−1(x)e =(2n) e 3 exp √ 2 2n 2 2 3 − 1 × − 2 3 Ai 2 z + o n (1 + o(1)) .
Now we start to estimate the integral in (1.4). We consider the interval of integration [0, ∞) (since the integral is even) and split it in the subintervals a), b), c) as in (2.8). And we split the interval b) in (2.8) into three subintervals: 2 1 +θ 1 b1) x ∈ 2n − n 3 ;2n − Mn3 ; 2 1 1 (2.12) b2) x ∈ 2n − Mn3 ;2n + Mn3 ; 2 1 1 +θ b3) x ∈ 2n + Mn3 ;2n + n 3 .
Thus,wehavesplittedx ∈ [0, ∞)onfivezones(seeFigure2.1).
Figure 2.1. Zones of R+, which gives different contribution to the integral, depending on p.
1 We recall, that θ is a fixed small number, such that 0 <θ< 6 and the constant M will be chosen depending on p. Making the change of variables √x = t in (2.11), we obtain for the integrals 2n along the interval a) in (2.8): ∞ p − 2 − 1 − − 2 x p(n 1)+ 2 pn p Ia = Hn−1(x)e dx =(2n) e 2 1 +θ 2n+n 3 ∞ 2 dt (2.13) × exp p(2n − 1)arccosh t − 2ntp t − 1+o(1) p , (t2 − 1) 2 θ− 2 1 3 1+ 4 n +n
24 A. I. APTEKAREV, J. S. DEHESA, P. SANCHEZ-MORENO,´ AND D. N. TULYAKOV and for the interval c) in (2.8):
− 1 +θ 2n n 3 p − 2 − 1 − 2 x p(n 1)+ 2 pn Ic = Hn−1(x)e dx =(2n) e 0 θ− 2 + 1− 1 n 3 n 4 p 2 2 dt × 1 − sin (2n − 1) arcsin 1 − t − 2nt 1 − t + o(1) p (1 − t2) 2 0 p(n−1)+ 1 −pn =(2n) 2 e
θ− 2 + − 1 3 n 1 4 n √ √ − − 2 − − 2 p 2p (2n 1) arcsin 1 t 2nt 1 t π dt × 2 sin − + o(1) p , 2 4 (1 − t2) 2 0 θ− 2 where n = o n 3 . Then we pass to the integrals along b)-(2.8). The idea to split interval b) into three subintervals (2.12) was because we are intending to use in the subintervals b1) and b3) in (2.12) the asymptotics of the Airy function from b)-(2.11); in b2)-(2.12) we shall use the explicit expression for the Airy function. Noticing that for n →∞, we have from definition of z in (2.9)
√ 2n 4 z dz − 3 ⇒ − √ ⇒ − √ z = 2 x x 2n 6 dx = 6 x 3 2 2n 2 2n
Thus,
1 2n −Mn3 p 2 −x2 Ib1 = Hn−1(x)e dx 1 +θ 2n−n 3
nθ p p n− 2 − 1 −pn 1 2 3 − p (2n) ( 3 ) 6 e 1+sin z 2 z 2 dz, 2 3 M
1 +θ 2n +n 3 p 2 −x2 Ib3 = Hn−1(x)e dx 1 2n+Mn3
nθ p n− 2 − 1 −pn −p−1 2 3 − p (2.14) (2n) ( 3 ) 6 e 2 exp − pz 2 z 2 dx; 3 M
ASYMPTOTICS OF Lp-NORMS OF HERMITE POLYNOMIALS AND RENYI´ ENTROPY 25 1 2n +Mn3 p 2 −x2 Ib2 = Hn−1(x)e dx 1 2n−Mn3 M √ p 3 p n− 2 − 1 −pn −p−1 2π 2 z 2 (2.15) (2n) ( 3 ) 6 e 2 √ Ai − dz. 3 2 2 −M The symbol means that the ratio of the left and right hand sides tends to unity. Now we can analyse the contributions of the various p-depending parts of the inte- gral of the left hand side of (2.10), when n →∞. We note, that all o(1) terms in our asymptotic analysis are differentiable, therefore they will not make contributions in our further estimates of the integrals. First, we notice that the integral part of Ia in the right hand side of (2.13) is exponentially small, and there exist constants α, c > 0, such that this integral is α 3 θ estimated as O n exp −cn 2 ,andforIa we have
p(n−1)+ 1 (2n) 2 α 3 θ I = O n exp −cn 2 . a 2penp Therefore this part is negligible for (2.10).
Second, we notice that the integral parts of Ib3 and Ib2 in the right hand side of (2.14) and (2.15), respectively, are O(1) and we have
p(n− 2 )− 1 −pn −p−1 Ib ,Ib =(2n) 3 6 e 2 O(1). 2 3 2−p θ 2 When p<2, the integral parts of Ic and Ib1 behave as O (1) and O n and we have p(n−1)+ 1 −pn Ic =(2n) 2 e O (1) , − pn − p n− 2 − 1 e θ 2 p I =(2n) ( 3 ) 6 O n 2 , b1 2 for 0 When p =2,thenbothintegralpartsIc and Ib1 have the same logarithmic rate of growth O(ln n). Computing the constant in O we obtain in a non-trivial way that ∞ 2 − 2 − 3 − 2 x 2n 2 2n Hn−1(x)e dx =(2n) e (ln(n)+O(1)) , for p =2. 0 Finally, for p>2 as we see from (2.13)-(2.15), the integral over b)-(2.8) domi- nates in (2.10). Thus taking M →∞,weobtain ∞ p 2 −x2 Hn−1(x)e dx −∞ √ p ∞ 3 − 2 − 1 − − 2π z 2 p(n 3 ) 6 pn p √ 2 − =(2n) e 2 3 Ai dz (1 + o(1)) ; p>2. −∞ 2 2 26 A. I. APTEKAREV, J. S. DEHESA, P. SANCHEZ-MORENO,´ AND D. N. TULYAKOV 3. R´enyi entropy of Rydberg oscillator states In this section Theorem 2.1 is applied to obtain the R´enyi entropy of the Ryd- berg states of the one-dimensional harmonic oscillator, described by the quantum- 1 2 mechanical potential V (x)= 2 x . It is in this energetic region where the transition from classical to quantum correspondence takes place. The physical solutions of the Schr¨odinger equation for the harmonic oscillator system (see e.g., [12]), are given 1 by the wavefunctions characterized by the energies En = n + 2 and the quantum probability densities 1 2 2 ρ˜ (x)=√ e−x H2(x) ≡ e−x H˜ 2(x), n πn!2n n n where H˜n(x) denotes the orthonormal Hermite polynomials of degree n. The degree n =0, 1, 2,... labels the energetic level. The entropic moments of these densities Wp[ρn] are expressed in terms of the entropic moments of the Hermite polynomials as 1 (3.1) Wp[˜ρn]= p Wp[ρn]. π 2 (n!)p2pn Thus, the entropic moments of the harmonic oscillator states are given by the en- tropic moments of the orthonormal Hermite polynomials. Consequently, according to equation (1.3) the R´enyi entropy of the harmonic oscillator for both ground and excited states is given by the concomitant R´enyi entropy of the involved orthonor- mal Hermite polynomials. Let us now consider the Rydberg states of the oscillator system; that is, the states with high and very high values of n. Then, taking into account equations (2.10) and (3.1), we obtain the asymptotic (n →∞)values ⎧ 1−p ⎪ cp (2n) 2 (1 + o(1)) ,p<2, ⎨⎪ −2 − 1 2π (2n) 2 (ln(n)+O(1)) ,p=2, (3.2) Wp[˜ρn−1]= ⎪ ⎪ Cp − 1 (p+1) ⎩ 2 (2n) 6 (1 + o(1)) ,p>2, (2π)p for the entropic moments of the Rydberg oscillator states. Finally, it is straightfor- ward to have the expressions for the R´enyi entropy of the Rydberg states as follows from Eqs. (3.2) and (1.3). 3 Figure 3.1 shows the values of the R´enyi entropy Rp[ρn]forp = 2 , p =2 and p = 3, as a function of n from n = 100 to n =1012. Notice that in all the cases the R´enyi entropy increases with n. This indicates that the spreading of these states increases with n. Moreover, for the values of p considered, after some initial intersections, the R´enyi entropy also increases when p decreases, for very large values of n,(n>107). This is also the observed behaviour when the R´enyi entropy is exactly calculated for low and moderate values of n (see e.g. [25]). Then, we can conclude that the observed intersections come from the differences between the asymptotic and the exact values of this quantity. 4. Conclusions In this work, we have shown that the R´enyi entropy of the one-dimensional harmonic oscillator is exactly equal to the R´enyi entropic integral of the involved orthonormal Hermite polynomials. Then, we have calculated the R´enyi entropy of ASYMPTOTICS OF Lp-NORMS OF HERMITE POLYNOMIALS AND RENYI´ ENTROPY 27 Figure 3.1. 3 R´enyi entropy Rp[ρn]forp = 2 (solid line), p =2 (dashed line) and p = 3 (dotted line) of the Rydberg oscillator states with n = 100 to n =1012. the highly excited states of the oscillator system by use of the asymptotics (n →∞) of the Lp-norms of the Hermite polynomials Hn(x) which control the corresponding wavefunctions. Remark that no recourse to the quasi-classical approximation has been done. The asymptotics of the Lp-norms of Hn(x) was determined by extending some sophisticated ideas and techniques extracted from the modern approximation theory [2, 29]. 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MR1963498 (2004h:01041) Keldysh Institute for Applied Mathematics, Russian Academy of Sciences and Mos- cow State University, Moscow, Russia E-mail address: [email protected] Departamento de F´ısica Atomica,´ Molecular y Nuclear, Universidad de Granada, Granada, Spain Instituto “Carlos I” de F´ısica Teorica´ y Computacional, Universidad de Granada, Granada, Spain E-mail address: [email protected] Departamento de Matematica´ Aplicada, Universidad de Granada, Granada, Spain Instituto “Carlos I” de F´ısica Teorica´ y Computacional, Universidad de Granada, Granada, Spain E-mail address: [email protected] Keldysh Institute for Applied Mathematics, Russian Academy of Sciences and Mos- cow State University, Moscow, Russia E-mail address: [email protected] Contemporary Mathematics Volume 578, 2012 http://dx.doi.org/10.1090/conm/578/11483 The next-order term for optimal Riesz and logarithmic energy asymptotics on the sphere J. S. Brauchart, D. P. Hardin, and E. B. Saff Dedicated to Paco Marcell´an on the occasion of his 60-th birthday. Abstract. We survey known results and present estimates and conjectures for the next-order term in the asymptotics of the optimal logarithmic energy and Riesz s-energy of N points on the unit sphere in Rd+1, d ≥ 1. The conjectures are based on analytic continuation assumptions (with respect to s)forthe coefficients in the asymptotic expansion (as N →∞)oftheoptimals-energy. 1. Introduction Let Sd denote the unit sphere in the Euclidean space Rd+1, d ≥ 1. The discrete logarithmic energy problem on Sd is concerned with investigating the properties of ∗ ∗ Sd N-point systems x1,...,xN on maximizing the product of all mutual pairwise Euclidean distances 2 M(x1,...,xN ):= |xj − xk| = |xj − xk| , j= k 1≤j (1) Elog(x1,...,xN ):= − log M(x1,...,xN ) 1 1 = log =2 log |xj − xk| |xj − xk| j= k 1≤j 2000 Mathematics Subject Classification. Primary 52A40; Secondary 31C20, 41A60. Key words and phrases. Dirichlet function, Logarithmic Energy, Riemann Zeta function, Riesz energy. The first author was a recipient of an APART-fellowship of the Austrian Academy of Sciences. Research conducted at the Center for Constructive Approximation at Vanderbilt University and the School of Mathematics and Statistics at the University of New South Wales. The research of the second and third authors was supported, in part, by the U. S. National Science Foundation under grants DMS-0808093 and DMS-1109266. c 2012 American Mathematical Society 31 32 J. S. BRAUCHART, D. P. HARDIN, AND E. B. SAFF We remark that in his list of “Mathematical problems for the next century” Smale [42, 43] posed as Problem #7 the challenge to design a fast (polynomial time) algorithm for generating “nearly” optimal logarithmic energy points on the unit sphere in R3 that satisfy 2 (2) Elog(x1,...,xN ) −Elog(S ; N) ≤ c log N for some universal constant c. This problem emerged from computational complexity theory (cf. Shub and Smale [41]). The right-hand side of (1) is referred to as the discrete logarithmic energy of the normalized counting measure μ[x1,...,xN ], which places the charge 1/N at each point x1,...,xN .Bythecontinuous logarithmic energy of a (Borel) probability measure μ supported on Sd we mean 1 I [μ]:= log d μ(x)dμ(y). log |x − y| Note that Ilog[μ]=+∞ for any discrete measure μ. Classical potential theory I yields# that log[μ] is uniquely minimized by the normalized surface area measure σd, M Sd Sd d σd =1,overtheclass ( ) of all (Borel) probability measures μ supported on Sd;thatis, ! " 1 1 (3) V (Sd):= inf I [μ] | μ ∈M(Sd) = I [σ ] = log + [ψ(d) − ψ(d/2)] , log log log d 2 2 where ψ(z):= Γ(z)/ Γ(z) is the digamma function. The last expression in (3) fol- lows, for example, from [14, Eq. (2.26)]. It is known that the minimum N-point logarithmic energy of Sd satisfies 1 log N c E (Sd; N) 1 log N c (4) V (Sd) − + 1 ≤ log ≤ V (Sd) − + 2 log 2 N N N 2 log d N N for some constants c1, c2 depending on d only. The lower bound follows from [49]. The upper bound follows from an averaging argument as in the proof of Theorem 1 in [27], which is based on equal area partitions [30]. These bounds give the correct form of the second-order term in the asymptotics of the minimum N-point logarithmic energy of S2. It should be mentioned that [38] also cites [29, p. 150] as a reference to a more general method to obtain lower bounds for minimum logarithmic energy valid over more general Riemann surfaces, which in the case of the 2-sphere also gives the lower bound in (4) (cf. [38, Sec. 3]). In a recent paper the first author improved the lower bound for higher-dimensional spheres. In [14, Lemma 1.1] it was shown that E (Sd; N) 1 log N C (5) log ≥ V (Sd) − − d + O (N −1−2ε/d),N→∞, N 2 log d N N ε where the positive constant Cd does not depend on N and is given by d/2 1 Γ(d)Γ(1+d/2−d/2) 1 1 C := V (Sd)+ + > 0; d log d 2d Γ(d/2) Γ(1 + d/2) 2 r r=1 the number ε satisfies 0 <ε <1(d even) or 0 <ε <1/2(d odd). Combining the upper bound in (4) and the lower bound in (5) we obtain the following asymptotic expansion of the minimum N-point logarithmic energy of Sd: 1 (6) E (Sd; N)=V (Sd) N 2 − N log N + O(N),N→∞. log log d THE NEXT-ORDER TERM 33 We remark that the minimum N-point logarithmic energy of the unit circle S is attained at the N-th roots of unity and (cf., for example, [15]) Elog(S; N)=−N log N, N ≥ 2. (In this case Vlog(S)=0.) In view of (6) it is tempting to ask if the following limit exists: 1 d d 2 1 (7) lim Elog(S ; N) − Vlog(S ) N + N log N =? N→∞ N d In particular, it is plausible and consistent with the lower bound (5) that this limit is negative for all d ≥ 2 if it exists. Indeed, in [38, 39] it is shown (also see below) that on the unit sphere in R3 (that is, d = 2) the square-bracketed expression in (7) can be bounded by negative quantities from below and above. 2 In formulating our Conjecture 4 for Elog(S ; N) (see Section 4), we will make heavy use of the observation that the discrete logarithmic energy is the limiting case (as s → 0) of the Riesz s-energy 1 1 Es(x1,...,xN ):= s =2 s , |xj − xk| |xj − xk| j= k 1≤j 2 wherewenotethatE0(A; N)=N − N, which is attained by any N-point config- uration on A.Furthermore,itisknown(cf.[12]) that % % d % (9) Es(A, N)% = Elog(A, N),N≥ 2. d s s=0+ As in the logarithmic case, the optimal continuous s-energy of Sd for −2 Sd Sd Sd and the s-capacity of is given by caps( ):=1/Vs( ). In Sections 3 and 4 we will discuss estimates and asymptotics for the optimal N-point s-energy on Sd. 2. The logarithmic energy on the unit sphere in R3 Let Δlog(N) N be the remainder term in 1 E (S2; N)=V (S2) N 2 − N log N +Δ (N) N. log log 2 log 34 J. S. BRAUCHART, D. P. HARDIN, AND E. B. SAFF Rakhmanov, Saff and Zhou [38, 39] proved the following estimates1 1 π −a b (11) lim inf Δlog(N) ≥− log 1 − e = −0.22553754 ..., N→∞ 2 2√ 1 π 3 π (12) lim sup Δlog(N) ≤− log + √ = −0.0469945 ..., N→∞ 2 2 4 3 where √ √ √ 2 2π √ √ 2π + 27 − 2π a:= √ 2π + 27 + 2π ,b:= √ √ , 27 2π + 27 + 2π and they stated the following conjecture: Conjecture 1([38]). There exist constants Clog,2 and Dlog,2, independent of N, such that (13) 1 1 E (S2; N)= − log 2 N 2 − N log N +C N +D log N +O(1),N→∞. log 2 2 log,2 log,2 The authors of [38] supported their conjecture by fitting the conjectured for- mula to the data obtained by high-precision computer experiments to find extremal configurations (N ≤ 200) and their logarithmic energy by minimizing the absolute -deviation. This leads to the following approximation2 1 1 1 E (S2; N) ≈ − log 2 N 2 − N log N − 0.052844 N +0.27644. log 2 2 (In [38] it is remarked that the logarithmic term was ignored in the fitting algorithm because it did not appear to be significant.) So far no conjecture for the precise value of Clog,2 in (13) has been given. Conjecture 4 in Section 4 fills this gap. Our approach for arriving at this conjecture is as follows: Since the logarithmic energy of an N-point configuration XN can be obtained by taking the derivative with respect to s of the Riesz s-energy of XN and letting s go to zero, we shall differentiate the asymptotic expansion of the minimal N-point Riesz s-energy of S2 (see (8)) and then let s go to zero from the right to derive plausible asymptotic formulas for the N-point logarithmic energy of S2.In this derivation there is an implicit interchange of differentiation and minimization which is formally justified by (9). In order to illustrate this approach, we begin with the simplest case, which is the unit circle S. Recently, we obtained in [15] the complete asymptotic expansion of the Riesz s-energy Ls(N)ofN-th roots of unity, which are the only (up to rotation) optimal energy points on the unit circle if s>−2, s =0,aswellasfor the logarithmic case. In fact, the expansion is valid for s ∈ C with s not zero or an odd positive integer: p 2 ζ(s) 2 L (N)=V (S) N 2 + N 1+s + α (s) ζ(s − 2n)N 1+s−2n s s (2π)s (2π)s n (14) n=1 −1+Re s−2p + Os,p(N )asN →∞(p =1, 2, 3,...), 1We remark that there is a typographical error in the sign of the second ratio in the upper bound in [38, Eq. (3.10)]. 2Note that the logarithmic energy in [38, 39] is precisely half of ours. THE NEXT-ORDER TERM 35 where ζ(s) is the classical Riemann zeta function. The coefficients αn(s), n ≥ 0, satisfy the generating function relation − ∞ sin πz s (−1)nB(s)(s/2) = α (s)z2n, |z| < 1,s∈ C,α(s)= 2n (2π)2n . πz n n (2n)! n=0 (α) Here, Bn (x) denotes the so-called generalized Bernoulli polynomial of degree n. In particular, one has the following recurrence relation for n ≥ 1(cf.Luke[31, p. 34f]) − n 1 2n − 1 B (15) B(2ρ)(ρ)=−2ρ 2m+2 B(2ρ) (ρ),B(2ρ)(ρ)=1, 2n 2m +1 2m +2 2n−2−2m 0 m=0 where B0 =1,B1 = −1/2, B2 =1/6, B3 = 0, . . . are the Bernoulli numbers. The constant Vs(S)istheRieszs-energy of the unit circle if −2 2−s Γ((1 − s)/2) (16) V (S):= √ ,s∈ C,s=1 , 3, 5,..., V := 1. s π Γ(1 − s/2) 0 Using properties of the digamma function and the Riemann zeta function, we obtain % % S % % d Vs( )% 1 % % = Vs(S) (ψ(1 − s/2) − ψ((1 − s)/2)) − log 2 % d s s→0 2 s→0 =0, % % % % d 2 ζ(s) 1+s % 2 ζ(s) 1+s ζ (s) − % s N % = s N log N + log(2π) % d s (2π) s→0 (2π) ζ(s) s→0 = −N log N. Moreover, using that the Riemann zeta function vanishes at negative even integers (0) ≥ and B2k (0) = 0 for k 1 (cf. (15)), we obtain % % d 2αn(s) 1+s−2n − % s N ζ(s 2n) % =0 forn =1, 2, 3,.... d s (2π) s→0 Thus, putting everything together, we arrive at % L % d s(N)% Elog(S; N)= % = −N log N, d s s→0 as we expected. A second fundamental observation is that the coefficients of the powers of N in the asymptotic expansion of Ls(N) can be extended to meromorphic functions on C. In this way one sees that the leading two terms “swap places” when moving s from the interval (−2, 1) to interval (1, 3). (Note that we assume that Ls(N)=N(N −1) at s = 0.) Thus, each term’s singularity at s = 1 can be avoided by moving away from the real line. In other words, the second term of the asymptotics of Ls(N)in the finite energy case s ∈ (−2, 1) is the analytic continuation of the leading term in the hyper-singular case s ∈ (1, 3) and vice versa. This is the motivation for trying the same ideas for the d-sphere in Section 4. 36 J. S. BRAUCHART, D. P. HARDIN, AND E. B. SAFF 3. Asymptotics of optimal Riesz s-energy 3.1. The dominant term. The leading term of the asymptotic expansion (as N →∞) of the maximal (if s<0) and minimal (if s>0) Riesz s-energy of a compact set A in Rp is well understood if A has finite s-energy (that is, positive s-capacity). For sets of vanishing s-capacity the leading term is rather well understood in the sense that the existence of the coefficient has been established for a large class of compact sets by the second and third authors, but the determination of this coefficient for s>d(except for one-dimensional sets) has to date remained a challenging open problem. The general theory for (the energy integral associated with) the continuous s-potential (s<0) covered in Bj¨orck [9] provides Frostman-type results, existence and uniqueness results for the equilibrium measure μs, and characterization of p the support of μs for general compact sets A in R . Of particular interest is the observation that the support of μs is concentrated in the extreme points of the convex hull of A if s<−1andfors<−2 any maximal distribution (there is no unique equilibrium measure anymore for s ≤−2) consists of no more than (p +1) point masses. The singular Riesz s-potential on Sd, 0 3This monotonicity holds more generally for any compact set A. THE NEXT-ORDER TERM 37 d We will denote this meromorphic extension with the same symbol Vs(S ). The hypersingular case s ≥ d. Since for s ≥ d the s-energy integral for every positive Borel probability measure supported on Sd is +∞, potential theory fails to work. (The boundary or exceptional case s = d can still be treated using a particular normalization of the energy integral and a limit process as s approaches d from below, see [16]). The dominant term of the asymptotic expansion of the minimal s-energy grows like N 2 log N in the boundary case s = d. The coefficient is also known. Theorem 2 (boundary case s = d,[27]). E Sd H Bd d( ; N) d( ) 1 Γ((√d +1)/2) (18) lim 2 = d = . N→∞ N log N Hd(S ) d π Γ(d/2) p Here and hereafter, Hd(·) denotes d-dimensional Hausdorff measure in R , p ≥ d, normalized such that a d-sided cube with side length 1 has Hd-measure equal to d d d 1. In particular, Hd(B ) denotes the volume of the unit ball in R while Hd(S ) denotes the surface area of the unit sphere in Rd+1. d In [27] the order of the growth rate of Es(S ; N) was established: there are constants C1,C2 > 0 such that 1+s/d d 1+s/d (19) C1 N ≤Es(S ; N) ≤ C2 N ,s>d≥ 2. d The second and third author [25] showed that the limit of the sequence Es(S ; N)/ N 1+s/d, indeed, exists. More generally, the following result holds, which has been referred to as the Poppy-seed Bagel Theorem because of its interpretation for dis- tributing points on a torus: Theorem 3([25, 11]). Let d ≥ 1 and A ⊂ Rp an infinite compact d-rectifiable set. Then for s>d & & 1+s/d s/d (20) lim Es(A; N) N = Cs,d [Hd(A)] , N→∞ where Cs,d is a finite positive constant (independent of A). By d-rectifiable set we mean the Lipschitz image of a bounded set in Rd. In particular, s/d E Sd d+1 s( ; N) Cs,d Γ 2 lim = = Cs,d ,s>d. N→∞ 1+s/d d s/d (d+1)/2 N [Hd(S )] 2π In [32,Thm.3.1]itisshownthatCs,1 =2ζ(s). For d ≥ 2 the precise value of Cs,d is not known. The significance (and difficulty of determining Cs,d) is deeply rooted in the connection to densest sphere packings. Let δN denote the best-packing distance of N-point configuration on Sd.Itisshownin[10]that 1/s 1/d d 1/d (21) lim [Cs,d] =1/C∞,d,C∞,d:= lim N δN =2[Δd/Hd(B )] , s→∞ N→∞ d where Δd is the largest sphere√ packing density in R , which is only known for three cases:√ Δ1 =1,Δ2 = π/ 12 (Thue in 1892 and L. Fejes T´oth [21]), and Δ3 = π/ 18 (Kepler conjecture proved by Hales [23]). d It is not difficult to see that the Epstein zeta function ζΛ(s) for a lattice Λ ⊂ R defined for s>dby −s ζΛ(s):= |x| 0= x∈Λ 38 J. S. BRAUCHART, D. P. HARDIN, AND E. B. SAFF yields an upper bound for Cs,d as we now show. Let Ω denote a fundamental parallelotope for Λ. For n ∈ N, the intersection Xn of Ω and the scaled lattice (1/n)Λ contains exactly N = nd points and, for s>d,wehave −s −s d+s Es(Xn)= |x − y| ≤ |x − y| = n ζΛ(s) x∈Xn, x= y∈Xn x∈Xn, x= y∈(1/n)Λ 1+s/d = N ζΛ(s). Referring to (20) with A = Ω, one obtains the following result. Proposition 1. For s>d, s/d (22) Cs,d ≤ min |Λ| ζΛ(s), Λ where the minimum is taken over all lattices Λ ⊂ Rd with covolume |Λ| > 0. In particular, as shown in [27], √ s/2 √ E S2 s/2 3 s( ,N) ( 3/2) ζΛ2 (s) Cs,2 ≤ ζΛ (s) and lim ≤ ,s>2, 2 2 N→∞ N 1+s/2 (4π)s/2 where√ Λ2 is the hexagonal lattice consisting of points of the form m(1, 0)+ n(1/2, 3/2) with m and n integers. For most values of d we do not expect equality to hold in (22); that is, we do not expect lattice packings to be optimal (especially for d large where it is expected that best packings are highly ‘disordered’ and far from being lattice arrangements, cf. [47]); however, recent results and conjectures of Cohn and Elkies [18], Cohn and Kumar [19] and Cohn, Kumar, and Sch¨urmann [20] suggest that equality holds in (22) for d =2, 4, 8 and 24 leading to the following: Conjecture . | |s/d 2 For d =2, 4, 8 and 24, Cs,d = Λd ζΛd (s) for s>d,where Λd denotes, respectively, the hexagonal lattice, D4, E8, and the Leech lattice. For d = 2, this conjecture appears in [27]. 3.2. The second-order term. The only known results so far are estimates d d 2 of the difference Es(S ; N) − Vs(S )N in the potential-theoretic regime. At the end of this section we present a lower bound for the optimal d-energy, the first hypersingular case. The average distance problem (s = −1) on a sphere was studied by Alexan- der [2, 4](d = 2) and Stolarsky [45](d ≥ 2, see citations therein for earlier work), later by Harman [26](d ≥ 2, correct order and signs of the bounds up to log N fac- tor in upper bound) and Beck [7](d ≥ 2, settled the correct order and signs of the bounds); generalized sums of distances (−2 THE NEXT-ORDER TERM 39 The method of the alternative proof in [38] was generalized in [27] leading to E Sd ≤ Sd 2 − 1+s/d s( ; N) Vs( ) N C2 N ,C2 > 0(0 40 J. S. BRAUCHART, D. P. HARDIN, AND E. B. SAFF Proposition 5. Let d ≥ 2 and s>d.Then,for(s − d)/2 not an integer, H Bd s/d E Sd ≤ d( ) 1+s/d s Sd 2 O 1+s/d−2/d s( ; N) d N + Vs( ) N + (N ). Hd(S )(1 − d/s) d The constant As,d in Proposition 4 and the corresponding constant in Propo- sition 5 provide lower and upper bounds for the constant Cs,d appearing in Propo- sition 2. Corollary 4. Let d ≥ 2 and s>d. Then for (s − d)/2 not an integer d s/d Cs,d Hd(B ) As,d ≤ ≤ , d s/d H Sd − [Hd(S )] d( )(1 d/s) where As,d is given in Proposition 4. d s/d For d = 2, the above bounds for Cs,d/[Hd(S )] reduce to & −s/2 −s/2 s/2 −s s/2 (23) 2 s [s/2 − 1] ≤ Cs,2/ [4π] ≤ 2 [s/ (s − 2)] . 4. Conjectures for the Riesz s-energy A straightforward generalization of the asymptotics for the unit circle (14) would be p d d 2 Cs,d 1+s/d 1+s/d−2n/d Es(S ; N)=Vs(S ) N + N + βn(s, d)N H Sd s/d (24) [ d( )] n=1 1+s/d−2p/d−2/d + Os,d,p(N ),N→∞ d for s not a pole of Vs(S ) given in (10) and Cs,d as defined in (20). The exceptional d cases are caused by the simple poles of Vs(S )asacomplexfunctionins.As s → d, one of the terms in the finite sum has to compensate for the pole of the N 2 term, thus introducing a logarithmic term (cf. the motivation for Conjecture 5 in Section 7). Recall that for even d there are only finitely many poles suggesting that there might only be finitely many exceptional cases for even d. It is possible that the higher-order terms (p ≥ 1) may not exist in this form;4 the presence of the N 2 and N 1+s/d is motivated by known results and the principle of analytic continuation. In the potential theoretic case −2 d d 2 Cs,d 1+s/d 1+s/d (25) Es(S ; N)=Vs(S ) N + N + o(N ) as N →∞, d s/d [Hd(S )] 4In fact, Michael Kiessling from Rutgers University pointed out that higher-order terms might not exist, instead one could have an oscillating term as N grows. Moreover, numerical results in Melnyk et al [33] suggest that for N fixed there might be an abrupt change of optimal configuration when an increasing s passes certain critical values that depend on N.Thisbehavior might also influence the existence of higher-order terms. THE NEXT-ORDER TERM 41 where, for s>d, the constant Cs,d is the same as that in (20).Furthermore,for | |s/d d =2, 4, 8 and 24, Cs,d = Λd ζΛd (s),whereΛd is as in Conjecture 2.For d =2, −2 − (27) ζΛ2 (s)=6ζ(s/2) L 3(s/2), Re s>2, into a product of the Riemann zeta function ζ(s)andthefirstnegativeprimitive Dirichlet L-Series 1 1 1 1 (28) L− (s):=1 − + − + −··· , Re s>1. 3 2s 4s 5s 7s 5This technique can be found in old papers addressing problems in solid state physics, cf. [22, p. 188f]. 42 J. S. BRAUCHART, D. P. HARDIN, AND E. B. SAFF 1400 1200 1000 800 600 400 200 0.2 0.4 0.6 0.8 1.0 Figure 1. The histogram (counting number of occurrences) and the empiric distribution function of the first 100000 distances of a local optimal 900-point 1-energy configuration. The vertical lines denote the hexagonal lattice distances adjusted such that the small- est distance coincidences with the best-packing distance. The Dirichlet L-series above can be also expressed in terms of the Hurwitz zeta function ζ(s, a) at rational values a,cf.[27]. That is, ∞ −s 1 (29) L− (s)=3 [ζ(s, 1/3) − ζ(s, 2/3)] ,ζ(s, a):= , Re s>1. 3 (k + a)s k=0 Remark. It is understood that ζΛ2 (s) in (26) is the meromorphic extension to C of the right-hand side of (27). Since ζ(s) is negative on the interval [−1, 1), has apoleats = 1, and is positive on (1, ∞) and the Dirichlet L-Series is positive on 6 the interval (−1, ∞) (cf., Eq. (27)), it follows that Cs,2 in (26) would be negative for −2 6 Note that L−3(1 − 2m)=0form =1, 2, 3, 4,...,cf.[50]. THE NEXT-ORDER TERM 43 Conjecture 4. For d =2, 4, 8,and24, 1 (30) E (Sd; N)=V (Sd) N 2 − N log N + C N + o(N),N→∞, log log d log,d where the constant of the N-term is given by 1 d (31) Clog,d = log(Hd(S )/|Λd|)+ζ (0). d Λd For the case d =2, (31) reduces to √ 1 2 π C = 2 log 2 + log + 3 log = −0.05560530494339251850 ...<0. log,2 2 3 Γ(1/3) Remark. We expect, more generally, that (30) holds for arbitrary d provided that C is differentiable at s = 0, in which case (31) becomes s,d % % d % d Clog,d = Cs,d% +(1/d)logHd(S ). d s s=0 Regarding the bounds (11) and (12), note that Clog,2 is closer to the upper bound given in (12), which gives rise to the question if the related argument can be improved to give the precise value. 4.1. The boundary case s = d. As with the unit circle, we expect to obtain the asymptotics of the optimal Riesz energy in the singular case from the corre- sponding asymptotics of the Riesz s-energy, s = d and s sufficiently close to d,by means of a limit process s → d. This approach leads to the next conjectures whose motivating analysis is given in Section 7. Conjecture 5. Let d ≥ 1.Then H Bd E Sd d( ) 2 2 O →∞ d( ; N)= d N log N + Cd,dN + (1),N , Hd(S ) where d Cs,d Cd,d = lim Vs(S )+ . s→d d s/d [Hd(S )] For d =2, this becomes √ 1 √ 3 C = γ − log(2 3π) + [γ (2/3) − γ (1/3)] 2,2 4 4π 1 1 = −0.08576841030090248365 ···< 0. Here, γ is the Euler-Mascheroni constant and γn(a) is the generalized Stieltjes n constant appearing in the coefficient γn(a)/n! of (1 − s) in the Laurent series expansion of the Hurwitz zeta function ζ(s, a) about s =1. 5. Numerical Results Rob Womersley from UNSW kindly provided numerical data, which we used to test our conjectures. For the logarithmic and the Coulomb cases (s =1)on S2 results for small numbers (N =4,...,500) and for large numbers of points (N =(n +1)2 points, N up to 22801) are given. The reader is cautioned that these numerical data represent approximate optimal energies which we denote by 2 2 Eˆlog(S ; N)orEˆs(S ; N). A general observation is the slow convergence of the sequence of s-energy values. 44 J. S. BRAUCHART, D. P. HARDIN, AND E. B. SAFF 0.00 0.01 0.02 0.03 0.04 0.05 10 20 50 100 200 500 Figure 2. The logarithmic case on S2, N =4, 5, 6 ...,500 points. The horizontal axis shows N on a logarithmic scale and the quan- tity (32) on the vertical axis. The horizontal line shows the con- jectured limit Clog,2. 0.01 0.02 0.03 0.04 0.05 10 50 100 500 1000 5000 1 104 Figure 3. The logarithmic case on S2 with N = 4, 9, 16 ...,10201 points. The same quantities are shown as in Fig- ure 2. THE NEXT-ORDER TERM 45 1.080 1.085 1.090 1.095 1.100 1.105 10 20 50 100 200 500 Figure 4. The Coulomb case (s =1)onS2, N =4, 5, 6 ...,500 points. The horizontal axis shows N on a logarithmic scale and the quantity (33) on the vertical√ axis. The horizontal line shows the conjectured limit C1,2/ 4π. 5.1. Logarithmic case. Figures 2 and 3 show the convergence to the conjec- tured coefficient of the N-term (see Conjecture 4) 1 (32) Eˆ (S2; N) − V (S2) N 2 − N log N /N. log log 2 The horizontal line indicates the value of Clog,2 given in Conjecture 4. 5.2. The Coulomb case s =1. Figures 4 and 5 show the convergence to the conjectured coefficient of the N 1+1/2-term (see Conjecture 3) ' ( 2 2 1+1/2 (33) Eˆ1(S ; N) − 1 × N /N . √ The horizontal line indicates the value of C1,2/ 4π. 5.3. The boundary case s = d =2. Figure 6 shows the convergence to the conjectured coefficient of the N 2-term (see Conjecture 5) 1 (34) Eˆ (S2; N) − N 2 log N /N 2. 2 4 The horizontal line indicates the value of C2,2. 6. Proofs In the following we set 2π(d+1)/2 ω := H (Sd)= . d d Γ((d +1)/2) 46 J. S. BRAUCHART, D. P. HARDIN, AND E. B. SAFF 1.085 1.090 1.095 1.100 1.105 10 50 100 500 1000 5000 1 104 Figure 5. The Coulomb case (s =1)onS2, N = 4, 9, 16 ...,22801 points. The same quantities are shown as in Fig- ure 4. 0.079 0.080 0.081 0.082 0.083 0.084 0.085 20 50 100 200 500 Figure 6. The boundary case (s =2)onS2, N =4, 5, 6 ...,500 points. The horizontal axis shows N on a linear scale and the quantity (34) on the vertical axis. The horizontal line shows the conjectured limit C2,2. Proof of lower bound in Proposition 3. This proof follows the first part of the proof of Theorem 3 in [27]. By an idea of Wagner, the (hyper)singular Riesz d-kernel 1/rd is approximated by the smaller continuous kernel 1/(ε + r2)d/2 THE NEXT-ORDER TERM 47 d 2 d/2 (ε>0). Then, for x, y ∈ S ,wehave1/(ε + |x − y| ) = Kε( x, y )where − −d/2 Kε(t):=(2 2t + ε) .NotethatKε is positive definite in the sense of Schoen- ∞ d berg [40]; that is, it has an expansion Kε(t)= n=0 an(ε) Pn (t) in terms of ultra- d λ λ spherical polynomials (normalized Gegenbauer polynomials Pn (t)=Cn (t)/Cn (1), where λ =(d − 1)/2) with positive coefficients an(ε)(n ≥ 1) giving rise to the estimates 1 E (X ) ≥ = K ( x , x ) − K ( x , x ) d N d/2 ε j k ε j j 2 (35) j=k ε + |xj − xk| j,k j 2 ≥ a0(ε) N − Kε(1) N, d where we used the positivity of Kε and P0 (t) = 1 (cf. Sec. 3 of [27]). Using the integral representation [1, Eq. 15.6.1] of the regularized Gauss hypergeometric function we obtain 1 ωd−1 −d/2 2 d/2−1 a0(ε)= (2 − 2t + ε) 1 − t d t ωd −1 1 −d/2 − ω − − 1 =(4+ε) d/2 2d−1 d 1 ud/2−1 (1 − u)d/2 1 1 − u d u ωd 0 1+ ε/4 −d/2 d−1 ωd−1 2 d/2,d/2 1 =(4+ε) 2 [Γ(d/2)] 2F˜1 ; . ωd d 1+ε/4 Assuming that ε<2, we apply the linear transformation [1, Eq. 15.8.11] with the understanding that ψ(d/2 − k)/ Γ(d/2 − k) is interpreted as (−1)k−d/2+1(k − d/2)! if d/2 − k is a non-positive integer: ∞ k 1 ωd−1 (d/2)k ε a0(ε)= − Γ(d/2) 2 ωd k!k!Γ(d/2 − k) 4 k=0 ε × log − 2 ψ(k +1)+ψ(d/2+k)+ψ(d/2 − k) 4 1 ωd−1 ε 1 − d/2,d/2 ε = − log 2F1 ; − 2 ωd 4 1 4 ∞ 1 ω − (d/2) ε k − d 1 Γ(d/2) k 2 ωd k!k!Γ(d/2 − k) 4 k=0 × [ψ(d/2+k)+ψ(d/2 − k) − 2 ψ(k +1)]. Note that the (non-regularized) Gauss hypergeometric function is a polynomial of degree d/2 − 1ifd is even and reduces to 1 if d = 2. Using the series representation [1, Eq. 15.2.1] we arrive at 1 ω1 ω1 1 a0(ε)= (− log ε)+ log 2 + O(ε)= log(4/ε)+O(ε)asε → 0ifd =2, 2 ω2 ω2 4 1 ωd−1 ωd−1 a0(ε)= (− log ε) − [ψ(d/2) − ψ(1) − log 2] + O(ε log(1/ε)) 2 ωd ωd as ε → 0ifd ≥ 3. 48 J. S. BRAUCHART, D. P. HARDIN, AND E. B. SAFF 2 −2/d −d/2 Using the substitution ε = a N and noting that Kε(1) = ε , there follows from (35) (one has O(N 2−2/d)asN →∞if d =2) $ 1 2 − 2 O d 4 N log N f(2; a) N + (N ),d=2, Ed(S ; N) ≥ 1 ωd−1 N 2 log N − f(d; a) N 2 + O(N 2−2/d log N),d≥ 3, d ωd as N →∞, where for each positive integer d the function ω − f(d; a):= d 1 [log a + ψ(d/2) − ψ(1) − log 2] + a−d,a>0, ωd ∗ −1/d has a single global minimum at a =[ωd−1/(dωd)] in the interval (0, ∞) with value 1 ω − 1 ω − f(d; a∗)= d 1 1 − log( d 1 )+d (ψ(d/2) − ψ(1) − log 2) . d ωd d ωd d d Let Fd:=ωd−1/(dωd). Then it is elementary to verify that Fd = Hd(B )/Hd(S ). ∗ It remains to show that f(d; a ) > 0. It is easy to see that Fd >Fd+2 > 0and F1 =1/π < 1andF2 =1/4 < 1. Thus, 1−log Fd > 0. Since the digamma function is strictly increasing, the expression ψ(d/2) − ψ(1) − log 2 > 0ford ≥ 4 making f(d; a∗) > 0ford ≥ 4. Direct computations show that f(1; a∗)=[1+log(π/8)] /π, f(2; a∗)=1/4,f(3; a∗)=2[7+log(3π/1024)] / (3π) are all positive, which completes the proof of the lower bound. We need the following auxiliary results for the upper bound in Proposition 3 . Let C(x,ρ) denote the spherical cap {y ∈ Sd : |y − x|≤ρ} = {y ∈ Sd : y, x ≥1 − ρ2/2}. Lemma 5. Let d ≥ 1. Then for x ∈ Sd and 0 <ρ≤ 2, the normalized surface area measure of the spherical cap C(x,ρ) is given by 1 ωd−1 d 1 − d/2,d/2 2 σd(C(x,ρ)) = ρ 2F1 ; ρ /4 , d ωd 1+d/2 where the Gauss hypergeometric function is a polynomial if d is even and reduces to 1 if d =2. Proof. Using the definition of the spherical cap, the Funk-Hecke formula (see [34]), and the substitution t =1− (ρ2/2)u, the surface area of a spherical cap can be written in terms of a hypergeometric function as follows 1 ωd−1 2 d/2−1 σd(C(x,ρ)) = d σd = 1 − t d t ωd − 2 C(x,ρ) 1 ρ /2 1 1 ω − − d/2−1 = d 1 ρd ud/2−1 (1 − u)1 1 1 − ρ2/4 u d u 2 ωd 0 1 ωd−1 d Γ(d/2) Γ(1) 1 − d/2,d/2 2 = ρ 2F1 ; ρ /4 2 ωd Γ(1 + d/2) 1+d/2 from which the result follows using properties of the Gamma function. Lemma 6. Let m be a positive integer. For z ∈ C \{−1, −2, −3,...} there holds m (z) (−z) − (1 − z) k m k [ψ(k + z) − ψ(k +1)]= m . k!(m − k)! m! m k=0 THE NEXT-ORDER TERM 49 Proof. Let fm(x) denote the sum for real x>0. Using the integral represen- tation [36, Eq. 2.2.4(20)] 1 α − β t t − − − d t = ψ(β +1) ψ(α +1), Re α, Re β> 1, 0 1 t we obtain 1 x−1 m 1 − t (x) (−x) − f (x)= g (x, t)dt, g (x, t):= k m k tk, m 1 − t m m k!(m − k)! 0 k=0 where the function gm(x, t) can be expressed as regularized Gauss hypergeometric functions Γ(1 + x) − g (x, t)=(−1)m F˜ m, x ; t m m! 2 1 1+x − m Γ(1 + x) − =(−1)m (1 − t) F˜ 1 m, 1+x; t . m! 2 1 1+x − m Substituting the series expansion of gm(x, t)wehave(forx − m not a negative integer) − Γ(1 + x) m 1 (1 − m) (1 + x) 1 f (x)= (−1)m k k tk − tk+x−1 d t m m! Γ(1 + x − m + k)k! k=0 0 − Γ(1 + x) m 1 (1 − m) (1 + x) 1 =(−1)m k k m! Γ(1 + x − m + k)k! k +1 k=0 − Γ(1 + x) m 1 (1 − m) (1 + x) 1 − (−1)m k k m! Γ(1 + x − m + k)k! k + x k=0 − 1 m (−m) Γ(k + x) 1 m 1 (1 − m) Γ(k + x) =(−1)m k − (−1)m k m!(−m) Γ(x − m + k)k! m! Γ(1 + x − m + k)k! k=1 k=0 1 Γ(x) m (−m) (x) =(−1)m k k m!(−m) Γ(x − m) (x − m) k! k=1 k − 1 Γ(x) m 1 (1 − m) (x) − (−1)m k k m! Γ(x +1− m) (x +1− m) k! k=0 k 1 Γ(x) − =(−1)m F m, x;1 − 1 m!(−m) Γ(x − m) 2 1 x − m 1 Γ(x) − − (−1)m F 1 m, x ;1 . m! Γ(x +1− m) 2 1 x +1− m 50 J. S. BRAUCHART, D. P. HARDIN, AND E. B. SAFF Using the Chu-Vandermonde Identity [1, Eq. 15.4.24] F −n, b;1 =(c − b) /(c) , 2 1 c n n we obtain (−m) − m 1 Γ(x) m − fm(x)= ( 1) − − − 1 m!( m) Γ(x m) (x m)m 1 Γ(x) (1 − m) − − (−1)m m 1 m! Γ(x +1− m) (x +1− m) m−1 1 Γ(x) Γ(x − m) 1 =(−1)m (−1)mm! − 1 − m!(−m) Γ(x − m) Γ(x) (−m) − m (1 − x) − m 1 Γ(x) 1 ( 1) m 1 =( 1) − = = , m! m Γ(x m) m! m (x)−m m! m where in the last line we used properties of the Pochhammer symbol (see, for example, [37, Appendix II.2]. Since fm(x) is, in fact, analytic in C with poles at negative integers due to the digamma function (the singularity at 0 can be removed), the identity (1 − x) f (x)= m m m! m canbeextendedtoC \{−1, −2, −3,...} by analytic continuation. Lemma 7. Let d ≥ 1.Forx ∈ Sd and 0 <ρ<2 −d ωd−1 1 ωd−1 |x − y| d σd(y)= (− log ρ) − [ψ(d/2) − ψ(1) − 2 log 2] Sd\C(x,ρ) ωd 2 ωd ∞ 1 ω − (1 − d/2) ρ 2m − d 1 m , 2 ω m! m 2 d m=1 where the series terminates after finitely many terms if d is even. Proof. Writing |x − y|−d =[2(1− t)]−d/2 with t = x, y and using the substitution 1 + t =2(1− ρ2/4)u the integral can be expressed as a hypergeometric function as in the proof of Lemma 5: −d |x − y| d σd(y) Sd\C(x,ρ) 1−ρ2/2 ω − − d/2−1 = d 1 [2 (1 − t)] d/2 1 − t2 d t ωd −1 1−ρ2/2 ω − − − = d 1 2−d/2 (1 − t) 1 (1 + t)d/2 1 d t ω − d 1 1 1 ω − d/2 − −1 = d 1 1 − ρ2/4 ud/2−1 (1 − u)1 1 1 − 1 − ρ2/4 u d u 2 ωd 0 1 ωd−1 2 d/2 1,d/2 2 = 1 − ρ /4 Γ(d/2) Γ(1) 2F˜1 ;1− ρ /4 . 2 ωd 1+d/2 THE NEXT-ORDER TERM 51 The linear transformation [1, Eq. 15.8.10] applied to the hypergeometric function above gives −d |x − y| d σd(y) Sd\C(x,ρ) ∞ 1 ω − d/2 (1) (d/2) = − d 1 1 − ρ2/4 k k (ρ/2)2k[2 log(ρ/2)+ψ(k+d/2)−ψ(k+1)] 2 ωd k!k! $k=0 ∞ 1 ω − d/2 ρ (d/2) ρ 2k = − d 1 1 − ρ2/4 2 log k 2 ωd 2 k! 2 k=0 ) ∞ (d/2) ρ 2k + k [ψ(k + d/2) − ψ(k +1)] k! 2 k=0 ∞ ω − ω − 1 ω − d/2 (d/2) = d 1 (− log ρ)+ d 1 log 2 − d 1 1 − ρ2/4 k ωd ωd 2 ωd k! k=0 ρ 2k × [ψ(k + d/2) − ψ(k +1)] . 2 The binomial expansion of the factor (1 − ρ2/4)d/2 is absolutely convergent for 0 ≤ ρ<2 as is the infinite series above. This gives ∞ d/2 (d/2) ρ 2k 1 − ρ2/4 k [ψ(k + d/2) − ψ(k +1)] k! 2 k=0 ∞ ∞ (d/2) (−d/2) ρ 2k+2n = k [ψ(k + d/2) − ψ(k +1)] n k! n! 2 k=0 n=0 ∞ ρ 2m = b (d) , m 2 m=0 where m (d/2) (−d/2) − b (d):= k m k [ψ(k + d/2) − ψ(k +1)]. m k!(m − k)! k=0 Then b0(d)=ψ(d/2) − ψ(1) and, by Lemma 6, (1 − d/2) b (d)= m ,m≥ 1. m m! m Thus, we obtain −d ωd−1 ωd−1 |x − y| d σd(y)= (− log ρ)+ log 2 d ω ω S \C(x,ρ) d $ d ) ∞ 1 ω − (1 − d/2) ρ 2m − d 1 ψ(d/2) − ψ(1) + m . 2 ω m! m 2 d m=1 Proof of upper bound in Proposition 3. A closer inspection of the sec- ondpartoftheproofofTheorem3in[27] gives almost (up to a log log N factor) the 52 J. S. BRAUCHART, D. P. HARDIN, AND E. B. SAFF ∗ ∗ ∗ correct order of the second term. Let XN =(x1,...,xN )beanoptimald-energy configuration on Sd.Forr>0consider *N Sd \ ∗ −1/d Dk(r):= C(xk,rN ),k=1,...,N, D(r):= Dk(r). k=1 ∗ Since XN is a minimal d-energy configuration, for each j =1,...,N the function 1 U (x):= , x ∈ Sd \ (X∗ \{x∗}), j | ∗ − |d N j k:k= j xk x ∗ −1/d attains its minimum at xj . By Lemma 7 and for ρ = rN (0 <ρ<2) we get 1 σ (D(r)) U (x ) ≤ U (x)dσ (x) ≤ d σ (x) d j j j d | ∗ − |d d D(r) Dk(r) x x $ k:k=j k ω − =(N − 1) d 1 − log(rN−1/d) ωd 1 ω − − d 1 [ψ(d/2) − ψ(1) − 2 log 2] 2 ωd ) ∞ −1/d 2m 1 ω − (1 − d/2) rN − d 1 m . 2 ω m! 2 d m=1 Hence, $ N N (N − 1) 1 ω − ω − E (Sd; N)= U (x∗) ≤ d 1 log N + d 1 (− log r) d j j σ (D(r)) d ω ω j=1 d d d 1 ω − − d 1 [ψ(d/2) − ψ(1) − 2 log 2] 2 ωd ) ∞ −1/d 2m 1 ω − (−d/2) 1 2 rN − d 1 m − . 2 ω (m − 1)! m d 2 d m=1 Subtracting off the dominant term of the asymptotic expansion, we obtain − − d 1 ωd−1 2 1 ωd−1 1 σd(D(r)) 1/N 2 Ed(S ; N) − N log N ≤ N log N d ωd d ωd σd(D(r)) 1 ω − N (N − 1) + d 1 − log rd d ωd σd(D(r)) 1 ω − N (N − 1) − d 1 [ψ(d/2) − ψ(1) − 2 log 2] 2 ωd σd(D(r)) ∞ −1/d 2m 1 ω − N (N − 1) (−d/2) 1 2 rN − d 1 m − . 2 ω σ (D(r)) (m − 1)! m d 2 d d m=1 Using Lemma 5 in the trivial bound (recall ρ = rN−1/d) (36) − ≥ − ∗ −1/d − 1 ωd−1 d 1 d/2,d/2 2 −2/d σd(D(r)) 1 Nσd(C(x1,rN )) = 1 r 2F1 ; r N /4 d ωd 1+d/2 THE NEXT-ORDER TERM 53 gives 1 ωd−1 d 1 − d/2,d/2 2 −2/d [1 − σd(D(r))] log N ≤ r log N 2F1 ; r N /4 . d ωd 1+d/2 Choosing rd =1/ log N,wearriveattheresult. Proof of Proposition 4. We follow the Proof of Proposition 3, now for the − −s/2 kernel Kε(s; t):=(2 2t + ε) which is positive definite in the sense of Schoen- ∞ d berg [40] with the expansion Kε(s; t)= n=0 an(s; ε) Pn (t). (The positivity of the coefficients an(s; ε) can be seen by applying Rodrigues formula (see [34]) and integration by parts n times.) We have 2 2 −s/2 Es(XN ) ≥ a0(s; ε) N − Kε(s;1)N = a0(s; ε) N − ε N, where the coefficient a0(s; ε) can be expressed in terms of a regularized Gauss hypergeometric function 1 ωd−1 −s/2 2 d/2−1 a0(s; ε)= (2 − 2t + ε) 1 − t d t ωd −1 1 −s/2 ω − − − 1 =2d−s−1 d 1 (1+ε/4) s/2 ud/2−1(1 − u)d/2 1 1 − u d u ωd 0 1+ε/4 d−s−1 ωd−1 −s/2 s/2,d/2 1 =2 Γ(d/2) Γ(d/2) (1 + ε/4) 2F˜1 ; . ωd d 1+ε/4 For (s − d)/2 not an integer we can do asymptotic analysis by applying the linear transformation [1, Eq. 15.8.5], that is 1 F˜ s/2,d/2; 2 1 d 1+ε/4 $ s/2 π (1 + ε/4) s/2, 1 − d + s/2 ε = F˜ ; − sin[π(d − s)/2] Γ(d − s/2) Γ(d/2) 2 1 1+(s − d)/2 4 & (d−s)/2 − (ε/4) (1 + ε/4) (1 + ε/4)d s/2 − Γ(s/2) Γ(d/2) ) d − s/2, 1 − s/2 ε × F˜ ; − , 2 1 1 − (s − d)/2 4 which, after simplifications and application of the last transformation in [1, Eqs. 15.8.1], yields 1 π s/2, 1 − d + s/2 ε a (s; ε)= V (Sd) F˜ ; − 0 s Γ((d − s)/2) sin[π(d − s)/2] 2 1 1+(s − d)/2 4 (d−s)/2 − − d−1−s ωd−1 Γ(d/2) π ε ˜ 1 d/2,d/2 − ε 2 2F1 − − ; . ωd Γ(s/2) sin[π(d − s)/2] 4 1 (s d)/2 4 54 J. S. BRAUCHART, D. P. HARDIN, AND E. B. SAFF Changing to Gauss hypergeometric functions shows that (d−s)/2 d−1−s ωd−1 Γ(d/2) π 1 ε a0(s; ε)=2 − − − ωd Γ(s/2) sin[π(s d)/2] Γ(1 (s d)/2) 4 1 − d/2,d/2 ε × F ; − 2 1 1 − (s − d)/2 4 Sd 1 1 π + Vs( ) − − − − Γ((d s)/2) Γ(1 (d s)/2) sin[π(d s)/2] s/2, 1 − d + s/2 ε × F ; − . 2 1 1+(s − d)/2 4 Application of the reflection formula for the gamma function [1, Eq.s 5.5.3] and the substitution ε/4=a2/(d−s)N −2/d gives − d−1−s ωd−1 Γ(d/2) Γ((s d)/2) −1+s/d a0(s; ε)=2 aN ωd Γ(s/2) 1 − d/2,d/2 − − (37) × F ; −a2/(d s)N 2/d 2 1 1 − (s − d)/2 s/2, 1 − d + s/2 − − + V (Sd) F ; −a2/(d s)N 2/d . s 2 1 1+(s − d)/2 Note that the first hypergeometric function above is a polynomial if d is even and reduces to 1 if d = 2. Hence, using the series expansion of a hypergeometric function, d 1+s/d d 2 1+s/d−2/d Es(S ; N) ≥ f(s, d; a) N + Vs(S ) N + O(N ),N→∞, where the function f(s, d; a):=ca− 2−sas/(s−d) with (cf. (18)) − − d−1−s ωd−1 Γ(d/2) Γ((s d)/2) d−1−s Γ((d +1)/2) Γ((s d)/2) c = cs,d =2 =2 √ ωd Γ(s/2) π Γ(s/2) has a unique maximum at a∗ =[2sc(s − d)/s](s−d)/d with value ∗ s s/d−1 −s s s/d As,d:=f(s, d; a )=c [2 c (s − d) /s] − 2 [2 c (s − d) /s] d s − d s/d d 1 Γ((d +1)/2) Γ(1 + (s − d)/2) s/d = 2s−d c = √ . s − d s s − d 2 π Γ(1 + s/2) For the proof of Proposition 5 we need the following auxiliary result. Lemma 8. Let d ≥ 1 and s>dand (s − d)/2 not an integer. For x ∈ Sd and 0 <ρ<2 we have d−s −s d 2 ωd−1 d−s |x − y| d σd(y)=Vs(S )+ (ρ/2) Sd\ s − d ω C(x,ρ) d 1 − d/2, (d − s)/2 ρ2 × F ; . 2 1 1 − (s − d)/2 4 THE NEXT-ORDER TERM 55 Proof. Similar as in the proof of Lemma 7 we obtain 1−ρ2/2 −s ωd−1 −s/2 2 d/2−1 |x − y| d σd(y)= [2 (1 − t)] 1 − t d t Sd\C(x,ρ) ωd −1 1−ρ2/2 ω − − − − = d 1 2−s/2 (1 − t)(d s)/2 1 (1 + t)d/2 1 d t ω − d 1 1 ω − d/2 − =2d−1−s d 1 1 − ρ2/4 ud/2−1 (1 − u)1 1 ωd 0 (d−s)/2−1 × 1 − 1 − ρ2/4 u d u d−1−s ωd−1 2 d/2 1+(s − d)/2,d/2 2 =2 1 − ρ /4 Γ(d/2) Γ(1) 2F˜1 ;1− ρ /4 , ωd 1+d/2 for (s − d)/2 not a positive integer we can apply the linear transformation [1, Eq. 15.8.4], $ ω − d/2 π Γ(d/2) Γ(1) =2d−1−s d 1 1 − ρ2/4 ωd sin[π(d − s)/2] Γ(d − s/2) Γ(1) 1+(s − d)/2,d/2 ρ2 × F˜ ; 2 1 1+(s − d)/2 4 ) 2 Γ(d/2) Γ(1) (d−s)/2 d − s/2, 1 ρ − ρ2/4 F˜ ; , Γ(1 + (s − d)/2) Γ(d/2) 2 1 1 − (s − d)/2 4 where the first regularized hypergeometric function can be evaluated using [1, Eq. 15.4.6] and the second can be transformed by the last linear transformation in [1, Eq. 15.8.1], d−1−s ωd−1 − 2 d/2 π =2 1 ρ /4 − $ ωd sin[π(d s)/2] Γ(d/2) −d/2 × 1 − ρ2/4 Γ(d − s/2)Γ(1+(s − d)/2) 1 (d−s)/2 −d/2 − ρ2/4 1 − ρ2/4 Γ(1 + (s − d)/2) ) 1 − d/2, (d − s)/2 ρ2 × F˜ ; 2 1 1 − (s − d)/2 4 ω − Γ(d/2) π 1 =2d−1−s d 1 ωd Γ(d − s/2) sin[π(d − s)/2] Γ(1 − (d − s)/2) ω − π 1 − − 2d−1−s d 1 (ρ/2)d s ω sin[π(d − s)/2] Γ(1+(s − d)/2) d 1 − d/2, (d − s)/2 ρ2 × F˜ ; , 2 1 1 − (s − d)/2 4 56 J. S. BRAUCHART, D. P. HARDIN, AND E. B. SAFF after changing to a non-regularized hypergeometric function and using the reflection formula for the gamma function, d−s ω − Γ(d/2) Γ((d − s)/2) 2 ω − − =2d−1−s d 1 + d 1 (ρ/2)d s ω Γ(d − s/2) s − d ω d d 1 − d/2, (d − s)/2 ρ2 × F ; . 2 1 1 − (s − d)/2 4 √ The substitution ωd−1/ωd =Γ((d +1)/2)/[ π Γ(d/2)] (cf. (18)) shows that the d first term is Vs(S ) by Eq. (10). The result follows. Proof of Proposition 5. We follow the proof of the upper bound in Propo- ∗ sition 3. Since XN is a minimal s-energy configuration, for each j =1,...,N the function 1 U (x):= , x ∈ Sd \ X∗ \{x∗} , j |x∗ − x|s N j k:k= j k attains its minimum at x∗. By Lemma 8 with ρ = rN−1/d (0 <ρ<2) we get j 1 σ (D(r)) U (x ) ≤ U (x)dσ (x) ≤ d σ (x) d j j j d |x∗ − x|s d D(r) Dk(r) k $ k:k=j d 1 ωd−1 d−s −1+s/d =(N − 1) Vs(S )+ r N s − d ωd ) 2 1 − d/2, (d − s)/2 r − × F ; N 2/d . 2 1 1 − (s − d)/2 4 Hence $ d 2 d 1 ωd−1 d−s −1+s/d σd(D(r)) Es(S ; N) ≤ N Vs(S )+ r N s − d ωd ) 2 1 − d/2, (d − s)/2 r − × F ; N 2/d 2 1 1 − (s − d)/2 4 1 ωd−1 d−s 1+s/d d 2 2+d−s 1+s/d−2/d = r N +Vs(S ) N +O(r N ). s − d ωd By relations (19) and (36), we have d 1 ωd−1 d d 2+d 1+s/d−2/d σd(D(r)) Es(S ; N) ≥ 1 − r Es(S ; N)+O(r N ). d ωd Note, that for d = 2 the hypergeometric function in (36) reduces to one and the O(·)-term above disappears. Hence d−s 1 ω − r E (Sd; N) ≤ d 1 N 1+s/d s 1 ωd−1 d s − d ωd 1 − r d ωd V (Sd) + s N 2 + O(r2+d−s N 1+s/d−2/d). 1 − 1 ωd−1 rd d ωd d−s d The function h(r)=r /(1 − cr )(wherec =(1/d)(ωd−1/ωd)) has a single minimum in the interval (0, ∞)atr∗ = c−1/d(1 − d/s)1/d with value h(r∗)= (s/d)c−1+s/d(1 − d/s)1−s/d,where1− crd = d/s > 0. The result follows. THE NEXT-ORDER TERM 57 7. Motivations for conjectures Motivation for Conjecture 4. Suppose Conjecture 3 holds. Proceeding formally, we obtain % % % | | s/d % d d d % d d 2 Λd 1+s/d % Elog(S ; N)= Es(S ; N) = Vs(S )N + ζ (s) N % % Λd % d s → + d s ωd s 0 s→0+ +Δ(Sd; N), where 1 |Λ | s/d Δ(Sd; N) = lim E (Sd; N) − V (Sd) N 2 − d ζ (s) N 1+s/d . s s Λd s→0+ s ωd Assuming the limit exists and that Δ(Sd; N)=o(N)asN →∞,wehave (38) % % d |Λ | s/d % E (Sd; N)= V (Sd)N 2 + d ζ (s) N 1+s/d % + o(N), log s Λd d s ωd % s→0+ 1 ζ (0) |Λ | = V (Sd)N 2 + ζ (0) N log N + ζ (0) + Λd log d log Λd Λd d d ωd × N + o(N), 1 1 |Λ | = V (Sd)N 2 − N log N + ζ (0) − log d N + o(N), log Λd d d ωd 1 = V (Sd)N 2 − N log N + C N + o(N)asN →∞, log d log,d where (cf. (3) and (10)) we used % % % − % d Sd % d d−s−1 Γ((d +1)√ /2) Γ((d s)/2) % Vs( )% = 2 − % d s s→0+ d s π Γ(d s/2) s→0+ 1 1 =log + [ψ(d) − ψ(d/2)] = V (Sd), 2 2 log − and also used the fact that ζΛ(0) = 1 holds for any√ lattice Λ (cf. [46]). Using Proposition 6 in the appendix and |Λ2| = 3/2 (for the hexagonal lattice with unit length edges), we obtain √ √ − 1 3 − log 3 − − 1 3 (39) Clog,2 = ζΛ (0) log =log(2π) 3logΓ(1/3) log 2 2 8π √ 4 2 8π 1 2 π = 2 log 2 + log + 3 log . 2 3 Γ(1/3) d Motivation for Conjecture 5. We first remark that Vs(S ) has a simple pole at s = d with a− V (Sd)= 1,d + A + O(|s − d|),s→ d, s s − d d where H Bd d d( ) − S − a 1,d:= Ress=d Vs( )= d d Hd(S ) 58 J. S. BRAUCHART, D. P. HARDIN, AND E. B. SAFF and d a−1,d 1 ωd−1 Ad:= lim Vs(S ) − = − (γ − 2log2+ψ(d/2)) . s→d s − d 2 ωd In addition to Conjecture 3, we further assume that Cs,d behaves, near s = d,as follows Cs,d (40) = b− /(s − d)+B + O(|s − d|),s→ d, d s/d 1,d d [Hd(S )] where Cs,d b−1,d b−1,d := −a−1,d and Bd:= lim − . s→d d s/d − [Hd(S )] s d Proceeding similarly as before and taking s → d,wehave d d 2 Cs,d 1+s/d d Es(S ; N)=Vs(S ) N + N +Δs(S ; N) d s/d [Hd(S )] Sd − a−1,d 2 a−1,d 2 − 1+s/d = Vs( ) − N + − N N s d s d Cs,d b−1,d 1+s/d d + − N +Δs(S ; N) d s/d − [Hd(S )] s d H Bd −→ 2 d( ) 2 2 Sd Ad N + d N log N + BdN +Δd( ; N), Hd(S ) d d 2 where we assume the limit Δd(S ; N)existsandthatΔd(S ; N)=o(N )asN → ∞. d s/d d s/d Multiplying both sides of (40) with Hd(S ) and expanding Hd(S ) about s = d, we obtain that ω − C = d 1 + B+ + O(|s − d|),s→ d, s,d s − d d where + ωd−1 Bd:= lim Cs,d − . s→d s − d Furthermore, the following relation holds between coefficient of the N 2-term and + Bd: + 1 ωd−1 1 ωd−1 Bd Cd,d:=Ad + Bd = − (γ − 2log2+ψ(d/2)) − log ωd + . 2 ωd d ωd ωd In the case d = 2 (with the help of Mathematica), we obtain 1−s 2 Cs,2 a−1,2 =Ress=2 = −1/2,b−1,2 =Ress=2 =1/2, 2 − s (4π)s/2 and 1−s 2 a−1,2 log 2 A2 = lim − = , s→2 2 − s s − 2 2 √ √ Cs,2 b−1,2 1 3 B2 = lim − = γ − log(8 3π) + [γ1(2/3) − γ1(1/3)] , s→2 (4π)s/2 s − 2 4 4π THE NEXT-ORDER TERM 59 where γ is the Euler-Mascheroni constant and γn(a) is the generalized Stieltjes constant appearing as the coefficient of (1 − s)n in the expansion of ζ(s, a)about s =1. Appendix A. Auxiliary results Proposition 6. 1 Γ(1/3) log 3 Γ(1/3) L− (0) = 1/3, L (0) = − log3+log = + 2 log √ , 3 −3 3 Γ(2/3) 6 √ 2π 3 3 Γ(2/3) log 3 ζ (0) = −1,ζ(0) = log √ + log =log(2π) − − 3logΓ(1/3). Λ2 Λ2 2π 2 Γ(1/3) 4 Proof. Note that one has the following identities (cf. [6, p. 264]) % % √ d % (41) ζ(0,a)=(1/2) − a, ζ(s, a)% =logΓ(a) − log 2π. d s s→0 By (29) and the last two relations L− (0) = ζ(0, 1/3) − ζ(0, 2/3) = (1/2) − (1/3) − (1/2) + (2/3) = 1/3, 3 % ! "% d −s − % L−3(0) = 3 [ζ(s, 1/3) ζ(s, 2/3)] % d s s→0 = − log 3 L−3(0) + log Γ(1/3) − log Γ(2/3). √ For the second relation for L−3(0) we used Γ(2/3) = 2π/[ 3Γ(1/3)]. By (27) and ζ(0) = −1/2, we get ζ (0) = 6 ζ(0) L−3(0) = 6 (−1/2) (1/3) = −1 Λ2 % % d % { − } − ζΛ2 (0) = 6 ζ(s/2) L 3(s/2) % =3ζ (0) L 3(0) + 3 ζ(0) L−3(0). d s s→0 Since ζ(s)=√ζ(s, 1), we derive from (41) the special values ζ(0) = −1/2and ζ (0) = − log 2π. This completes the proof. References [1] Digital Library of Mathematical Functions. 2010-05-07. 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School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia E-mail address: [email protected] Center for Constructive Approximation, Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240 E-mail address: [email protected] Center for Constructive Approximation, Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240 E-mail address: [email protected] Contemporary Mathematics Volume 578, 2012 http://dx.doi.org/10.1090/conm/578/11475 Spectral transformations of hermitian linear functionals M. J. Cantero, L. Moral, and L. Vel´azquez Abstract. In this paper we obtain a difference equation relating the Schur parameters of two quasi-definite hermitian linear functionals connected by an arbitrary rational modification. Independently of the intrinsic interest of this relation, it also provides a way to generate new families of orthogonal polyno- mials from known ones. We illustrate these results with an example. 1. Introduction Spectral transformations constitute a way to study connections between linear functionals which provide a method to build new families of orthogonal polynomials from known ones. The spectral transformations on the real line have been exten- sively analyzed, while the analogue on the unit circle has received less attention (see for instance [1]-[13],[15]-[22],[25],[28],[29]). Despite the intensive study of spectral transformations on the unit circle carried out during the last decade by F. Marcell´an and coworkers, some known results on the real line remain as open questions on the unit circle. For instance, it is known that the Christoffel and Geronimus transformations generate all the linear spectral transformations on the line, but this is merely a conjecture on the circle. This paper can be understood as a step forward in the analysis of spectral transformations on the unit circle. It constitutes a continuation of [4], where we consider a general kind of spectral transformations: arbitrary polynomial modifica- tions of hermitian linear functionals. Here we focus our attention in a more general problem: the study of rational modifications of hermitian linear functionals. As we will see, such modifications are linear spectral transformations which extend those ones generated by the Christoffel and Geronimus transformations. Indeed, we will show that rational modifications of hermitian functionals connect quasi- definite hermitian functionals which cannot be related by any chain of Christoffel and Geronimus transformations. This proves that the Christoffel and Geronimus transformations cannot generate all the linear spectral transformations between quasi-definite functionals, and we only could expect them to generate the linear spectral transformations between positive definite functionals. 2000 Mathematics Subject Classification. Primary 42C05. The work of the authors was partially supported by the research projects MTM2008-06689- C02-01 and MTM2011-28952-C02-01 from the Ministry of Science and Innovation of Spain and the European Regional Development Fund (ERDF), and by Project E-64 of Diputaci´on General de Arag´on (Spain). c 2012 American Mathematical Society 63 64 M. J. CANTERO, L. MORAL, AND L. VELAZQUEZ´ Along the paper we will deal with hermitian linear functionals u defined in the complex vector space of Laurent polynomials Λ := C[z,z−1], i.e., u ∈ Λ,where Λ denotes the algebraic dual of Λ. Actually, we will only consider quasi-definite hermitian linear functionals u, that is, there exists a sequence of monic orthogonal polynomials (ϕn) satisfying (i) ϕn ∈ Pn \ Pn−1, (ii) (ϕn,ϕm)u = lnδn,m,ln =0, Pn being the vector subspace of polynomials with complex coefficients whose degree −1 is not greater than n,and(f,g)u = u[f(z)g(z )] being the corresponding sesquilin- ear functional in Λ. We will also use the notation P := C[z] for the subspace of polynomials of any degree. The sequence (ϕn) satisfies the forward recurrence relation (see [26, 14, 23]) ∗ (1.1) ϕn(z)=zϕn−1(z)+ϕn(0)ϕn−1(z),n=1, 2 ..., with ϕ0(z)=1and|ϕn(0)| =1for n ≥ 1. Notice that these polynomials also satisfy ∗n n −1 a backward recurrence relation. We define the ∗n operator as p (z)=z p(z ), ∗ ∗n p ∈ Pn, and we write p = p when deg p = n. Applying ∗n to (1.1), we get the equivalent recurrence ∗ ∗ (1.2) ϕn(z)=ϕn(0)zϕn−1(z)+ϕn−1(z),n=1, 2 .... The values ϕn(0) are called the Schur parameters or reflection coefficients of the hermitian linear functional u. In this paper we propose a method to study the equality uL = vM,where L, M ∈ Λ, and uL[f]=u[Lf], f ∈ Λ. We will say that two functionals related by the above equality are rational modifications of each other. Assuming that u, v are quasi-definite, we will characterize a rational modification uL = vM by a relation between the Schur parameters of u and v. In a previous paper ([4]),wehavestudiedtheparticularcaseu = vM.A priori, it seems that the equality uL = vM could be analyzed through a pair of simpler problems: w = uL, w = vM. Nevertheless, this is not always possible because the intermediate functional w could be non-quasi-definite, or even worse, non-hermitian (see the example in Section 4). For convenience, we summarize in Table 1 the notation which will be used for the study of rational modifications. Spectral transformations are defined in terms of the Stieltjes and Carath´eodory ‘functions’, S(z)andF (z), which can be defined for any functional u ∈ Λ (not necessarily hermitian) as the formal power series given in Table 1. In the hermitian −1 − case μ−n = μn and both functions are related by F (z)=2z S∗(z) μ0,where S∗(z):=S(1/z). If the Stieltjes functions S(z)andT (z)ofu and v are related by A(z)S(z)+B(z) (1.3) T (z)= , A,B,D ∈ P, D(z) we say that u → v is a linear spectral transformation (see for instance [29]and references therein). When u, v are hermitian this relation is usually expressed in terms of the more significant Carath´eodory functions (see [22]). Indeed, the con- nection between Stieltjes and Carath´eodory functions shows that in the hermitian case (1.3) is equivalent to a similar linear relation with rational coefficients between the Carath´eodory functions F (z)andG(z)ofu and v. SPECTRAL TRANSFORMATIONS OF HERMITIAN LINEAR FUNCTIONALS 65 Table 1. Notation related to the rational modification uL = vM Functionals u v OP ϕn ψn Schur parameters an = ϕn(0) bn = ψn(0) −n −n en := (ϕn,ϕn)u = u[ϕnz ] εn := (ψn,ψn)v = v[ψnz ] , , n −| |2 n −| |2 ‘Squared norm’ = e0 k=1 1 ak , = ε0 k=1 1 bk , e0 := u[1] ε0 := v[1] n n Moments μn := u[z ] νn := v[z ] −(n+1) −(n+1) Stieltjes S(z):= n≥0 μnz T (z):= n≥0 νnz n n Carath´eodory F (z):=μ0 +2 n≥1 μ−nz G(z):=ν0 +2 n≥1 ν−nz Given a functional u, the equation uL = vM, L, M ∈ Λ, can have in general different solutions v. Let us see that any such a solution defines a linear spectral transformation u → v. The simplest case u(z − α)=v, α ∈ C, yields T (z)= (z − α)S(z) − μ0. An iteration of this result proves that the equality uA = v, A ∈ P, implies that T = AS + B, B ∈ P. Hence, rewriting the equality uL = v, L ∈ Λ, as uA = zrv, A ∈ P, we find that it leads to T = LS +L˜, L˜ ∈ Λ. Finally, the relation uL = vM needs LS + L˜ = MT + M˜ , L,˜ M˜ ∈ Λ, so that T =(LS + N)/M , N ∈ Λ, which is equivalent to (1.3). When u, v are hermitian this can be restated −1 using Carath´eodory functions as G =(L∗F +N˜)/M∗, N˜ =2z N∗ +μ0L∗ −ν0M∗. A rational modification of special interest is u(z−α)(z−1−α)=v, α ∈ C∗.This defines for each α a unique linear spectral transformation u → v which preserves the hermitian character, known as a Christoffel transformation (see [12]and[10, Chapter 2]). On the contrary, for a fixed α, the above rational modification defines a one-parameter family of linear spectral transformations v → u between hermitian functionals: given a hermitian v and a particular hermitian solution u = u0,the rest of hermitian solutions have the form u = u0 +mδ(z −α)+mδ(z −1/α), m ∈ C. Any of these relations v → u is known as a Geronimus transformation (see [9]and [10, Chapter 3]). Therefore, rational modifications uL = vM cover any spectral transformation generated by the Christoffel and Geronimus ones. An example of a rational modification which cannot be generated by compo- sition of Christoffel and Geronimus transformations is u(αz + α)=v(βz + β), α, β ∈ C∗. Any such a composition leads to a linear spectral transformation (1.3) with A and D of even degree, while this example leads to A and D of degree one. Rational modifications can be considered as an extension of the spectral transfor- mations generated by the Christoffel and Geronimus ones. This extension allows linear spectral transformations (1.3) with odd degree for the polynomials A and D. Further, we will see in Section 4 that the example u(αz + α)=v(βz + β)provides linear spectral transformations which, not only preserve the hermitian character, but can connect quasi-definite functionals. Therefore, the extended class of linear spectral transformations underlying the rational modifications becomes of interest for the study and generation of new families of orthogonal polynomials. 66 M. J. CANTERO, L. MORAL, AND L. VELAZQUEZ´ The content of the paper is structured in the following way: Section 2 is devoted to study of a “minimal” representation of the equality uL = vM,sincetheLaurent polynomials L, M are not unique. Section 3 includes a characterization of the rational modification in terms of a relation between the two families of orthogonal polynomials. It also includes a more important characterization in terms of a difference equation for the sequences of Schur parameters of u and v. The study of the algorithm that provides these relations is the main objective of Section 4, including and example which illustrates the preceding results with a non-trivial case which shows that a rational modification cannot be split, in general, in two polynomial modifications. 2. Minimality In what follows we study the equality uL = vM where L, M are Laurent polynomials. It is easy to check that, if this equation is satisfied for some Laurent polynomials L, M, it is also satisfied for any multiple of them. However, the possibility that the functionals u, v can involve Dirac deltas does not allow us to divide in general the equality uL = vM by a common factor of L and M (see [21]). Thus, a priori, it is not obvious the existence of a “minimal” representation for the equation uL = vM. In this section we look for the simplest equation, i.e., we will try to eliminate the ambiguity in the Laurent polynomials L, M finding the simplest ones with the best properties. We will give a canonical expression of uL = vM determined by a (unique) pair of self-reciprocal polynomials (A, B), with L = Az−p, M = Bz−q (p, q ∈ Z) such that the equality uAz−p = vBz−q will be minimal, i.e., the polynomials A, B have minimum degree. Besides we will see that, even in the non-minimal case, the degrees of A and B have the same parity. The following result will be useful for our purposes. Proposition 2.1. Let u be hermitian quasi-definite linear functional and L ∈ Λ. Then, uL =0iffL =0. q j Proof. Suppose L(z)= λj z . The hypothesis implies j=−p q k uL[z ]= λj μj+k, ∀k ∈ Z, j=−p k where μk = u[z ]. Taking k = −q,...,p, ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ··· λq 0 μ0 μ−p−q ⎜ ⎟ ⎜ ⎟ ⎝ ··· ··· ··· ⎠ . . ⎝ . ⎠ = ⎝.⎠ . μ ··· μ p+q 0 λ−p 0 Since u is hermitian quasi-definite, λj =0,j= −p,...,q. The following result is a direct consequence of the previous proposition. Proposition 2.2. Let u, v be hermitian quasi-definite linear functionals. The correspondence L ↔ M between Laurent polynomials L, M given by uL = vM is one-to-one. Next, we define the set of polynomials where we will introduce the minimality. SPECTRAL TRANSFORMATIONS OF HERMITIAN LINEAR FUNCTIONALS 67 Definition 2.3. For hermitian linear functionals u, v,wedefinethesets • Pˆ = {P ∈ P | P (0) =0 }∪{0}. • A = {A ∈ Pˆ |∃B ∈ Pˆ ∃p, q ∈ Z s.t. uAz−p = vBz−q } • I = {(A, B) ∈ Pˆ × Pˆ |∃p, q ∈ Z s.t. uAz−p = vBz−q} The order P1 −p −q1 −p −q2 uA1z = vB1z ,uA2z = vB2z . We suppose deg A2 ≤ deg A1 and we develop the Euclidean algorithm Ai = Ai+1Qi + Ai+2, with deg Ai+2 < deg Ai+1, for i =1, 2 ...,m− 2, and Am =gcd(A1,A2). −p −qk By induction, if uAkz = vBkz ,(k =1,...,i+1),wehave −p −qi uAiz = vBiz −p = u(Ai+1Qi + Ai+2)z −qi+1 −p = vBi+1Qiz + uAi+2z . From it we deduce −p −qi −qi+1 uAi+1z = v(Biz − Bi+1Qiz ) −qi+2 =: vBi+2z , with Bi+2(0) = 0. As a consequence, for all Ak obtained from the Euclidean −p −q algorithm, there exists Bk ∈ Pˆ such that uAkz = vBkz k , and this equality −p −qm is also satisfied for k = m.FromAm(0) =0, uAmz = vBmz , we find that Am ∈ A.SinceAm divides A1 and A2, this contradicts the hypothesis of minimality. ∗ Then, there exists λ ∈ C such that A2 = λA1. This means that A has a unique minimal polynomial up to constant factors. Since any element of A is divided by the essentially unique minimal polynomial, such a minimal polynomial plays the role of a generator of A. Corollary 2.5. The set I is monogenic, i.e., there exists (A, B) ∈ I such that I = Pˆ(A, B). Proof. Let A be a generator of A. From Proposition 2.2, there exists a unique B ∈ Pˆ such that uAz−p = vBz−q.Let(A,˜ B˜) ∈ I, i.e., uAz˜ −p = vBz˜ −q .Since A˜ = PA with P ∈ Pˆ,wehave uAz˜ −p = vBz˜ −q = uP Az−p = vPBz−q. 68 M. J. CANTERO, L. MORAL, AND L. VELAZQUEZ´ Thus Bz˜ −q = PBz−q by Proposition 2.2, and q = q, B˜ = PB.Inotherwords, (A,˜ B˜)=P (A, B). If (A, B)isageneratorofI, we will say that uAz−p = vBz−q is a minimal expression of uL = vM. Proposition 2.6. Let u, v be hermitian quasi-definite linear functionals and (A, B)ageneratorofI. Then, (A∗,B∗)=λ(A, B)forsomeλ ∈ C∗. Proof. Since u, v are hermitian, from uAz−p = vBz−q, we deduce p q uA∗z = vB∗z , uA∗z− deg A+p = vB∗z− deg B+q, and, then, (A∗,B∗) ∈ I, i.e., (A∗,B∗)=P (A, B)forsomeP ∈ Pˆ. Having in mind that deg A∗ =degA,degB∗ =degB, we deduce that deg P =0. Remark 2.7. Proposition 2.6 implies that we can give a unique (up to real fac- tors) generator of I such that (A, B)=(A∗,B∗), that we consider as the canonical generator. This result ensures that we can avoid all the non-self-reciprocal single factors in the equation uL = vM: first, if a non-self-reciprocal single factor is not common to A˜ and B˜, the equation uAz˜ −p = vBz˜ −q is impossible to satisfy; second, the common non-self-reciprocal single factors of A˜ and B˜ can be canceled. That is, although in general we cannot divide A,˜ B˜ by a common single factor, this can be done when the factor is not self-reciprocal. Proposition 2.8. Let u, v be hermitian quasi-definite linear functionals such that I = {(0, 0)}. Then, for all (A, B) ∈ I,degA ≡ deg B (mod 2). Proof. Let (A, B)beageneratorofI, r =degA, s =degB.Wehave uAz−p = vBz−q,uA∗z−r+p = vB∗z−s+q. Hence, vAB∗z−s+q = uAA∗z−r+p = vAB∗z−s+q = vA∗Bz−r−q+2p. Since A∗B = AB∗, it follows that r − s =2(p − q). If (A,˜ B˜) is not a generator, then (A,˜ B˜)=P (A, B)with(A, B) generator and P ∈ Pˆ. As a consequence, deg A˜ and deg B˜ have the same parity. 3. Characterization The equality uL = vM can be trivially characterized in terms the moments of the functionals u and v. This approach is too naive to obtain useful information about the relations between u and v, since it requires the use of determinantal expressions with increasing size. When u and v are hermitian quasi-definite we will give a characterization of uL = vM in terms of the OP related to u, v, and, subsequently, in terms of the corresponding sequences of Schur parameters. This last characterization will lead to the construction of a simple algorithm that provides the rational relation between u and v. The next lemma will be useful for our purposes. SPECTRAL TRANSFORMATIONS OF HERMITIAN LINEAR FUNCTIONALS 69 Lemma 3.1. Let u be a quasi-definite hermitian linear functional, (ϕn)n≥0 the corresponding sequence of monic OP, and w ∈ Λ . For any r ∈ Z+ the following statements are equivalent: −r i) There exists A ∈ Pr such that w = uAz . ∗ −n ∀ ≥ ii) w[ϕn+r]=w[ϕn+rz ]=0, n 1. Moreover, A is unique. r j Proof. i)⇒ ii) Let A(z)= αj z . Then, j=0 r −r+j w[ϕn+r]= αj u[ϕn+rz ]=0, ∀n ≥ 1. j=0 r ∗ −n ∗ −n−r+j ∀ ≥ w[ϕn+rz ]= αj u[ϕn+rz ]=0, n 1. j=0 ⇒ { }r ∪{ } ∪{ ∗ −j } ii) i) Let B = ϕj j=0 ϕj+r j≥1 ϕj+rz j≥1. Obviously B is a basis r −r j of Λ. We consider the functional uAz − w,whereA(z)= αj z ∈ Pr.From j=0 the hypothesis, we have r −r j−r (uAz − w)[ϕn+r]= αj u[ϕn+rz ]=0, ∀n ≥ 1, j=0 r −r − ∗ −n ∗ −n−r+j ∀ ≥ (uAz w)[ϕn+rz ]= αj u[ϕn+rz ]=0, n 1. j=0 ∀ ∈ P −r − { } ∪{ ∗ −j } In other words, A r, uAz w vanishes on ϕj+r j≥1 ϕj+rz j≥1.There- −r −r fore, uAz − w = 0 in Λ is equivalent to uAz − w =0 inPr, i.e., −r w[ϕj ]=u[Az ϕj ],j=0,...,r. These conditions yield a system of linear equations such that the matrix of coeffi- cients is lower triangular ⎧ ⎪ −r ⎪ w [ϕr]=α0u ϕ rz ⎨ −r −r−1 w [ϕr−1]=α0u ϕr−1z + α1u ϕr−1z ⎪ ··· ⎩⎪ −r w [ϕ0]=α0u ϕ0z + ···+ αru [ϕ0] , −j with diagonal entries ej = u[ϕj z ] =0( j =0,...,r), which ensures the existence and uniqueness for α0,α1,...,αr. Corollary 3.2. Under the conditions of Lemma 3.1, −n i) A(0) =0 ⇔ w[ϕn+rz ] =0. ⇔ ∗ ii) deg A = r w[ϕn+r] =0. r −n −n−r+j Proof. The identity w[ϕn+rz ]= αj u[ϕn+rz ]=α0en+r proves j=0 i). The proof of ii) is analogous. 70 M. J. CANTERO, L. MORAL, AND L. VELAZQUEZ´ Lemma 3.3. Let v be a hermitian quasi-definite linear functional and (ψn)n≥0 its sequence of OP. Then, for all t ∈ N ⊥ { j }t ∪{ j ∗ }t−1 tP n+2t i) B1,n = z ψn+t j=0 z ψn+t j=0 is a basis of (z n−1) . ⊥ { j }t ∪{ j ∗ }t t+1P n+2t ii) B2,n = z ψn+t j=1 z ψn+t j=0 is a basis of z n−1 . ⊥m Here, S represents the orthogonal complement of the vector subspace S ⊂ Pm in Pm Proof. Since j −k v[z ψn+tz ]=0,j=0,...,t, k = t,...,n+ t − 1, j ∗ −k − − v[z ψn+tz ]=0,j=0,...,t 1,k= t,...,n+ t 1, ⊥ t n+2t it follows that B1,n ⊂ (z Pn−1) . Let P ∈ span(B1,n), i.e., there exist C ∈ Pt, D ∈ Pt−1 such that P = Cψn+t + ∗ ∗ Dψn+t.Sincegcd(ψn+t,ψn+t) = 1 this expression for P is unique, which proves the linear independence of B1,n. Taking into account that B1,n =2t +1 = ⊥ ⊥ t n+2t t n+2t dim (z Pn−1) , we find that B1,n is a basis of (z Pn−1) . The proof of ii) is similar. If n =0,thenB1,0 and B2,0 are bases of P2t. Theorem 3.4. Let u, v be hermitian quasi-definite linear functionals, (ϕn)and (ψn) respectively the related sequences of monic OP, B ∈ Ps and t ∈ N, such that r =2t − s ≥ 0. The following statements are equivalent: −r −t i) There exists A ∈ Pr such that uAz = vBz . ii) There exist Cn,Fn ∈ Pt and Dn,En ∈ Pt−1 such that ∗ ∀ ≥ Bϕn+r = Cnψn+t + Dnψn+t, n 0, ∗ ∗ ∀ ≥ Bϕn+r = zEnψn+t + Fnψn+t, n 0. ↔ Moreover, the correspondence A ((Cn,Dn,En,Fn))n≥0 is bijective. −k −k−r+t Proof. i)⇒ ii) Let A∈Pr satisfying i). Since v[Bϕn+rz ]=u[Aϕn+rz ] ⊥ t n+r+s vanishes for t ≤ k ≤ n + t − 1, it follows that Bϕn+r ∈ (z Pn−1) = ⊥ tP n+2t ∗ (z n−1) . By Lemma 3.3, we can conclude that Bϕn+r = Cnψn+t + Dnψn+t, with Cn,Dn unique, for all n ≥ 0. ∗ −k ∗ −k−r+t ≤ ≤ Analogously, from v[Bϕn+rz ]=u[Aϕn+rz ]=0ift +1 k n + t, ⊥ ∗ ∈ t+1P n+2t ∗ ∗ it follows that Bϕn+r z n−1 , i.e., Bϕn+r = zEnψn+t + Fnψn+t with En,Fn unique, for all n ≥ 0. −t ii)⇒ i) Let Cn,Dn,En and Fn satisfying ii). We write w := vBz . Then, −t ∗ −t w[ϕn+r]=v[Bϕn+rz ]=v[(Cnψn+t + Dnψn+t)z ], ∗ −n ∗ −n−t ∗ −n−t w[ϕn+rz ]=v[Bϕn+rz ]=v[(zEnψn+t + Fnψn+t)z ], and both expressions are zero for all n ≥ 1. −r According to Lemma 3.1, there exists a unique A ∈ Pr such that w = uAz . Corollary 3.5. Under the conditions of Theorem 3.4, the following state- ments are satisfied: i) deg B = s ⇔ deg Cn = t. ii) B(0) =0 ⇔ Fn(0) =0. iii) deg A = r ⇔ deg Fn = t. iv) A(0) =0 ⇔ Cn(0) =0. SPECTRAL TRANSFORMATIONS OF HERMITIAN LINEAR FUNCTIONALS 71 Proof. i) and ii) follows by identifying the leading and the constant term in both hand sides of ii) in Theorem 3.4. r t Let A(z)=αrz + ···+ α0, Fn(z)=fnz + ···. Then, ∗ −t ∗ −t v[Bϕn+rz ]=u[Aϕn+rz ]=αren+r ∗ −t = v[(zEnψn+t + Fnψn+t)z ]=fnεn+t, from which iii) follows. Analogously, −n−t −n−r v[Bϕn+rz ]=u[Aϕn+rz ]=α0en+r ∗ −n−t = u[(Cnψn+t + Dnψn+t)z ]=Cn(0)εn+t, which yields iv). We can express these results in a compact way, using the following matrix notation: ϕn ψn a) Φn := ∗ , Ψn := ∗ . ϕn ψn b) The matrices zan zbn Sn = , Tn = anz 1 bnz 1 are known as transfer matrices of u and v, respectively. Then, the recurrence relations (1.1), (1.2) for (ϕn)and(ψn) can be written as Φn+1 = Sn+1Φn, Ψn+1 = Tn+1Ψn. c) Given the polynomials Cn,Fn ∈ Pt, Dn,En ∈ Pt−1 the relations ii) of Theorem 3.4 can be expressed together as BΦn+r = CnΨn+t, where C is the polynomial matrix n Cn Dn Cn = . zEn Fn Remark 3.6. Without loss of generality, we can suppose that the polynomial B of Theorem 3.4 satisfies B(0) =0anddeg B = s. With this notation, Theorem 3.4 can be enunciated in the following way: Theorem 3.7. Let u, v be hermitian quasi-definite linear functionals, (ϕn), (ψn) the related sequences of monic OP, B ∈ P such that deg B = s, B(0) =0and t ∈ N such that r =2t − s ≥ 0. The following statements are equivalent: − − i) There exists A ∈ P such that deg A = r, A(0) =0and uAz r = vBz t. Cn Dn ii) There exist matrix polynomials Cn = ,withdegCn =degFn = zEn Fn t, Cn(0) =0, Fn(0) =0and Dn,En ∈ Pt−1, such that BΦn+r = CnΨn+t, ∀n ≥ 0. Moreover the correspondence A ↔ (Cn)n≥0 is bijective. 72 M. J. CANTERO, L. MORAL, AND L. VELAZQUEZ´ For convenience, in what follows we will use the notation Cn Dn Mt = Cn = ;degCn =degFn = t, Cn(0),Fn(0) =0 ,Dn,En ∈ Pt−1 . zEn Fn Note that the matrices in Mt have non-zero determinant. Besides, according to Remark 2.7, all the non-self-reciprocal factors of uL = vM can be cancelled, then the polynomial B can be chosen such that B = B∗,which ∗ ∗ ∗t−1 necessarily implies A = A and Fn = Cn, En = Dn . In this case, the structure of the matrices Cn is remarkable. If we denote 01 × ∗ J = , we will say that the matrix C∈P2 2 is ∗-J invariant if C t = JCJ. 10 t ∗ It is immediate that if B = B , the matrices Cn are ∗-J -invariant. We denote by Jt the set of matrices of Mt that are ∗-J -invariant. Using this terminology, Theorem 3.7 can be enunciated for B = B∗ in the following way: Theorem 3.8. Let u, v be hermitian quasi-definite linear functionals, (ϕn), (ψn) their corresponding sequences of monic OP, B ∈ P such that deg B = s, B∗ = B, and t ∈ N such that r =2t − s ≥ 0. The following statements are equivalent: i) There exists A ∈ P such that deg A = r, A∗ = A and uAz−r = vBz−t. ii) There exist matrices Cn ∈ Jt such that BΦn+r = CnΨn+t, ∀n ≥ 0. Moreover, the correspondence A ↔ (Cn)n≥0 is bijective. Thus, we have characterized the relation uL = vM in terms of the OP related to u and v, respectively. In that follows, we will search for a characterization which does not require explicitly the OP, but only the Schur parameters of u and v. Theorem 3.9. Let u, v be hermitian quasi-definite linear functionals, (ϕn), (ψn) their corresponding sequences of monic OP, B ∈ P such that deg B = s, B(0) =0and t ∈ N such that r =2t − s ≥ 0. The following statements are equivalent: i) There exists A ∈ P, such that deg A = r, A(0) =0and uAz−r = vBz−t. ii) There exist polynomial matrices Cn ∈ Mt such that (3.1) BΦr = C0Ψt, (3.2) Sn+rCn−1 = CnTn+t, ∀n ≥ 1. Moreover, the correspondence A ↔ (Cn)n≥0 is bijective. Proof. By Theorem 3.7, there exist matrices Cn ∈ Mt such that BΦn+r = CnΨn+t, ∀n ≥ 0. s t t Let B(z)=βsz +···+β0. Then, Cn(z)=βsz +···+cn, Fn(z)=fnz +···+β0. t−1 If we also write Dn(0) = dn, En(z)=ηnz + ···,fromBΦn+r = CnΨn+t,we obtain (3.3) β0an+r = cnbn+t + dn, (3.4) βsan+r = ηn + bn+tfn, by identifying the constant term and the leading coefficients in both hand sides. The equality BΦn+r = CnΨn+t,forn = 0 gives (3.1), while for n ≥ 1, BΦn+r = BSn+rΦn+r−1 = Sn+rCn−1Ψn+t−1 SPECTRAL TRANSFORMATIONS OF HERMITIAN LINEAR FUNCTIONALS 73 = CnΨn+t = CnTn+tΨn+t−1. This yields (3.5) Sn+rCn−1Ψn+t−1 = CnTn+tΨn+t−1, which we can write as follows 1 an+r Cn−1 zDn−1 zψn+t−1 (3.6) ∗ = an+r 1 En−1 Fn−1 ψn+t Cn Dn 1 bn+t zψn+t−1 = ∗ . zEn Fn bn+t 1 ψn+t s Using Cn(z)=βz + ···, (3.3) and (3.4), the first equation of (3.6) ∗ (Cn−1 + an+rEn−1)zψn+t−1 +(an+rFn−1 + zDn−1)ψn−t+1 = ∗ =(Cn + bn+tDn)zψn+t−1 +(bn+tCn + Dn)ψn+t−1 ∗ ∈ P admits a representation Xnzψn+t−1 + zYnψn+t−1 = 0, with Xn,Yn t−1.Since ∗ ∀ ≥ gcd(ψn+t−1,ψn+t−1) = 1, we conclude that Xn = Yn =0, n 1. Analogously, using Fn(0) = β0, (3.3) and (3.4), we see that the second equation of (3.6) ∗ (an+rCn−1 + En−1)zψn+t−1 +(Fn−1 + an+rzDn−1)ψn−t+1 = ∗ =(bn+tFn + zEn)zψn+t−1 +(Fn + bn+tzEn)ψn+t−1 ˜ ˜ ∗ ˜ ˜ ∈ P admits a representation Xnzψn+t−1 + zYnψn+t−1 = 0, with Xn, Yn t−1. Then, X˜n = Y˜n =0, ∀n ≥ 1. As a consequence, from (3.5) we deduce that Sn+rCn−1 = CnTn+t, ∀n ≥ 1. ii)⇒i) It is enough to prove that BΦn+r = CnΨn+t, ∀n ≥ 0, by Theorem 3.4. If it is true for some n ≥ 0, then BΦn+r+1 = BSn+r+1Φn+r = Sn+r+1CnΨn+t S C T −1 C = n+r+1 n n+t+1Ψn+t+1 = n+1Ψn+t+1. Since (3.1) is satisfied by hypothesis, then BΦn+r = CnΨn+t, ∀n ≥ 0. Theorem 3.9 also admits the following version, based on the simplification of non-self-reciprocal factors. Theorem 3.10. Let u, v be hermitian quasi-definite linear functionals, (ϕn), ∗ (ψn) their corresponding SMOP, B ∈ P such that deg B = s, B = B,andt ∈ N such that r =2t − s ≥ 0. The following statements are equivalent: i) There exists A ∈ P such that deg A = r, A∗ = A and uAz−r = vBz−t. ii) There exist matrices Cn ∈ Jt such that BΦr = C0Ψt, Sn+rCn−1 = CnTn+t, ∀n ≥ 1. According to the previous results it is possible to exchange the roles of u, A and v, B. So, Theorems 3.7 and 3.9 allow us to enunciate the following result: Given u, v hermitian quasi-definite linear functionals with (ϕn), (ψn) their cor- responding sequences of monic OP, A ∈ P such that deg A = r, A(0) =0 ,andt ∈ N such that s =2t − r ≥ 0, the following statements are equivalent: 74 M. J. CANTERO, L. MORAL, AND L. VELAZQUEZ´ i) There exists B ∈ P such that deg B = s, B(0) =0 and uAz−t = vBz−s.(Note that uAz−t = vBz−s ⇔ uAz−r = vBz−t). ii) There exist matrices Dn ∈ Mt such that AΨn+s = DnΦn+t, ∀n ≥ 0. iii) There exist matrices Dn ∈ Mt such that AΨs = D0Φt, Tn+sDn−1 = DnSn+t, ∀n ≥ 1. The relations between the matrices Cn and Dn will provide us an expression of det(Cn) in terms of the polynomials A and B. Previously, we will need the following propositions: Proposition 3.11. Let u, v be hermitian quasi-definite linear functionals, (ϕn), (ψn) their corresponding sequences of monic OP, B ∈ P such that deg B = s, B(0) =0,and t ∈ N such that r =2t − s. The following statements are equivalent: i) There exists A ∈ P, such that deg A = r, A(0) =0and uAz−r = vBz−t. ii) For every n0 ≥ 0, there exist matrices Cn ∈ Mt such that BΦn+r = CnΨn+t, ∀n ≥ n0. ↔ C Moreover, the correspondence A ( n)n≥n0 is bijective. Proof. According to Theorem 3.10, it will be enough to prove that there exist Cj ∈ Mt such that BΦj+r = Cj Ψj+t,forj =0,...,n0 − 1. Let j ∈{1,...,n0} such that BΦj+r = Cj Ψj+t. From the recurrence, BSj+rΦj+r−1 = Cj Tj+rΨj+t−1. Let us define Cj−1 by S−1 C T C BΦj+r−1 = j+r j j+rΨj+t−1 =: j−1Ψj+t−1. It will be enough to prove that Cj−1 ∈ Mt. A straightforward calculation gives 1 c11 c12 C − j 1 = 2 , (1 −|aj+r| )z c21 c22 with c11 = z[Cj − aj+rzEj + bj+t(Dj − aj+rFj )], c12 = bj+t(Cj − aj+rzEj )+Dj − aj+rFj , 2 c21 = z [zEj − aj+rCj + bj+t(Fj − aj+rDj )], c22 = z[bj+t(zEj − aj+rCj )+Fj − aj+rDj )]. Using equations (3.3), (3.4) of Theorem 3.9, it is easy to check that Cj−1 ∈ Mt. Proposition 3.12. Let u, v be hermitian quasi-definite linear functionals such that I = {(0, 0)},andlet(ϕn), (ψn) be their corresponding sequences of monic OP. ∈ I C Then, (A, B) is minimal if and only if the entries of n0 are coprime for some ≥ n0 0, i.e., gcd(Cn0 ,Dn0 ,En0 ,Fn0 )=1. Proof. Note that (A, B) ∈ I implies the existence of matrices (Cn)n≥0 such that BΦn = CnΨn+t, Cn ∈ Mt, ∀n ≥ 0, with r =degA, s =degB, r + s =2t. 0 ∈ P C ∈ M − Let P be a divisor of Cn0 , Dn0 , En0 , Fn0 and n0 t deg P such that C C0 n0 = P n0 . From (3.2), Sn+rCn−1 = CnTn+t SPECTRAL TRANSFORMATIONS OF HERMITIAN LINEAR FUNCTIONALS 75 for n = n0 and n = n0 + 1, it follows that P is divisor of Cn,Dn,En,Fn, ∀n ≥ 0. C0 C ∀ ≥ Let n = n/P , n 0. From C C0 BΦn+r = nΨn+t = P nΨn+t, ∗ since gcd(ϕn+r,ϕn+r)=1,thenP is a divisor of B.Thus,B = PB0, with B0(0) =0.Let s0 =degB0 = s − deg P , t0 = t − deg P , r0 =2t0 − s0.Now,we can write C0 ∀ ≥ B0Φn+r = nΨn+t, n 0, or, equivalently, C0 ∀ ≥ B0Φn+deg P +r0 = nΨn+deg P +t0 , n 0. Proposition 3.11 implies the existence of A0 ∈ P,degA0 = r0, A0(0) =0with −r0 −t0 −r0−deg P −t0−deg P −t −r uA0z = vB0z , hence uP A0z = vBz = vBz = uAz . Proposition 2.2 implies that A = PA0 and (A, B)=P (A0,B0). Then (A, B)is minimal iff P ∈ C∗. Theorem 3.13. The matrices Cn of Theorem 3.7 satisfy en+r det(Cn)= AB. εn+t ∗ ∗ Proof. a) Suppose (A, B)=(A ,B ) minimal, then Cn ∈ Jt.FromBΦn+r = CnΨn+t, we find that BC+Φ = C+C Ψ =det(C )Ψ , n n+ r n n n+t n n+t ∗ C −Dn ∗ C+ n∗ C where n = t−1 is the adjoint matrix of n.Sincegcd(ψn+t,ψn+t)= −zDn Cn 1, B must be divisor of det(Cn), ∀n ≥ 0. ≥ { − } ˜ C ˜∗ X C+ We set n = k, with k max 0,r t , A =det( k)/B = A and k = k . Note that deg A˜ = r. Now, the equation BΦk+r = CkΨk+t can be written A˜Ψk+t = XkΦk+r. We define the matrices X ek+r εn+t C+ ∀ ≥ n = n , n k, εk+t en+r which belong to Jt. Considering that, ∀n ≥ k, en+r εn+t−1 en+r εk+t det(Cn)= det(Cn−1)= det(Ck), en+r−1 εn+t εn+t ek+1 i.e., en+r εk+t (3.7) det(Cn)= AB,˜ εn+t ek+t it follows that A˜Ψn+t = XnΦn+r, ∀n ≥ k. According to Proposition 3.11, there exists B˜ = B˜∗,degB˜ = s such that uAz˜ −t = vBz˜ −s, i.e., (A,˜ B˜) ∈ I. Since deg A =degA˜ = r,degB =degB˜ = s, then, there exists λ ∈ R∗ such that (A,˜ B˜)=λ(A, B). −n−r −n−t On the other hand, from u[Aϕn+rz ]=v[Bϕn+rz ]=v[(Cnψn+t + ∗ −n−t Dnψn+t)z ], we have 76 M. J. CANTERO, L. MORAL, AND L. VELAZQUEZ´ (3.8) A(0)en+r = Cn(0)εn+r. Taking z = 0 in (3.7), en+r εk+t Bn(0)Cn(0) = λA(0)B(0), εn+t ek+r e which, together with (3.8), gives λ = k+r , i.e., εk+t en+r det(Cn)= AB. εn+t b) Let (A, B), (A,˜ B˜) ∈ I with (A, B)=(A∗,B∗) minimal. Then, (A,˜ B˜)= P (A, B)forsomeP ∈ Pˆ. On the other hand, BΦn+r = CnΨn+t, Cn ∈ Jt, ∀n ≥ 0, and B˜Φn+deg P +r = C˜n+deg P Ψn+deg P +r, Cn+deg P ∈ Mt, ∀n ≥ 0. Since B˜ = PB, it follows that B˜Φn+r = P CnΨn+t, ∀n ≥ deg P, and, consequently, C˜n = P Cn, i.e., 2 en+r 2 en+r det(C˜n)=P det(Cn)= P AB = A˜B,˜ εn+t εn+t for all n ≥ deg P . Proposition 3.11 extends the result ∀n ≥ 0. Proposition 3.11, Proposition 3.12 and Theorem 3.13 have remarkable conse- quences. Proposition 3.11 shows that, if the relation BΦn+r = CnΨn+t is satisfied for all n ≥ n0, then it is also satisfied for all n ≥ 0. Thus, the equality BΦn+r = CnΨn+t for n = n0 implies the equality for the lower degrees n AΨm+s = DmΦm+t for all m ≥ k + t − s, and, by Proposition 3.11, ∀m ≥ 0. Moreover, it is immediate to check that CnDm = ABI2, I2 being the identity matrix of order two. We can summarize all these results in the following table: SPECTRAL TRANSFORMATIONS OF HERMITIAN LINEAR FUNCTIONALS 77 Table 2 uAz−r = vBz−t uAz−r = vBz−s Th3.4 BΦn+r = CnΨn+t (n ≥ 0) AΨn+s = DnΨn+t (n ≥ 0) Th3.9 BΦr = C0Ψt, AΨs = D0Ψt, Sn+rCn−1 = CnTn+t, (n ≥ 1) Tn+sDn−1 = DnSn+t, (n ≥ 1) en+r εn+s Th3.13 det(Cn)= AB (n ≥ 0) det(Dn)= AB, (n ≥ 0) εn+t en+t 4. Applications Theorems 3.9 and 3.13 give rise to an algorithm which, known one of the functionals and one of the polynomials in the equality uAz−p = vBz−q, allows to obtain the other functional and the other polynomial. Given v hermitian quasi- definite and B = B∗ with s =degB,andr ∈ N such that r + s =2t, t ∈ N, we will obtain the functional u and the polynomial A, such that uAz−r = vBz−t. The equation (3.1) of Theorem 3.9 provides the initial condition to calculate C0: from BΦr = C0Ψt a unique matrix C0 is determined in terms of the free parameters a1,...,ar.IfC0 ∈ Jt (i.e., if C0(0) = 0), we continue the process using the equation (3.2) of Theorem 3.9 Sn+rCn−1 = CnTn−t, or, equivalently, 1 an+r Cn−1 zDn−1 Cn Dn 1 bn+t (4.1) ∗t−1 ∗ = ∗t−1 ∗ , an+r 1 Dn−1 Cn−1 zDn Cn bn+t 1 s t where, analogously to Theorem 3.9 we will write B = βz + ···+ β, Cn = βz + t−1 ···+ cn, Dn = ηnz + ···+ dn. Setting z = 0 in (4.1), 1 a c − 0 c d 1 b n+r n 1 = n n n+t . an+r 1 ηn−1 β 0 β bn+t 1 Identifying the coefficient (2, 1) in both sides of the above equation yields − βbn+t ηn−1 (4.2) an+r = cn−1 whenever cn−1 =0.If |an+r| =1,then Cn ∈ Jt can be obtained from (3.2), Theorem 3.9, or (4.1). Finally, Theorem 3.13 gives ε det(C ) A = t 0 = A∗, er B up to a factor which can be identified with the election of e0. As a conclusion the algorithm can be implemented as follows, • Data: v, B, r. • Free parameters: a1,...,ar. 78 M. J. CANTERO, L. MORAL, AND L. VELAZQUEZ´ εt det(C0) • Initial conditions: BΦr = C0Ψt. Obtention of C0 and A = . er B • Algorithm: For n ≥ 1 If cn−1 =0, Determination of an+r according to (4.2). If |an+r| =1, C S C T −1 ∈ J ⇒ Determination of n = n+r n−1 n−t t cn =0. If v, A, s are known it is easy to check that the free parameters are a1,...,at. The rest of the algorithm is analogous, in terms of the matrices Dn. 4.1. Example. In what follows we will illustrate the preceding results with a non-trivial example. We will consider the rational modification of degree one u(αz +¯α)=v(βz + β¯),α,β∈ C∗, where v is the Lebesgue functional. It is very well known that the sequence of n orthogonal polynomials is ψn(z)=z , for all n ≥ 0, and εn =1,bn = 0, for all n ≥ 1. Our data are the functional v and the polynomial A(z)=αz + α (α =0). We try to find the quasi-definite hermitian functional u and the polynomial B −1 −1 (deg B = 1) such that uAz = vBz . The parameter a1 remains free. We start with the initial conditions AΨ = D Φ . 1 0 1 αz + γ0 δ0 We denote D0 = . Then, zδ0 γ0z + α (αz + α)z =(αz + γ0)(z + a1)+δ0(a1z +1), and 1 − 1 − (4.3) γ0 = 2 (α αa1),δ0 = 2 a1(αa1 α). 1 −|a1| 1 −|a1| 2 Keeping in mind that 1 −|a1| = e1/e0 and taking e0 = 1 by rescaling α if it is necessary, (4.3) reds as α − αa1 a1(αa1 − α) γ0 = ,δ0 = . e1 e1 Now, we can calculate the polynomial B according to Theorem 3.13, i.e., D − | |2 ε1 det( 0)=(αz + γ0)(γ0z + α) z δ0 = (αz + α)(βz + β). e1 2 Identifying the coefficients of z ,weobtainβ = e1γ0,thatis, (4.4) β = α − α a1. The assumption β = 0 implies that D0 ∈ Jt. A straightforward calculation shows that − C D+ βz + αe1 a1(α αa1) βz + c0 d0 0 = e1 0 = =: . a1(α − αa1)zαe1z + β zd0 c0z + β SPECTRAL TRANSFORMATIONS OF HERMITIAN LINEAR FUNCTIONALS 79 Setting z = 0 in (4.1), 1 a c − 0 c d n+1 n 1 = n n , an+1 1 dn−1 β 0 β and we obtain the recurrence (4.5) cn−1 + an+1dn−1 = cn, (4.6) βan+1 = dn, (4.7) cn−1an+1 + dn−1 =0. From (4.5) and (4.7), 2 en+1 cn = cn−1(1 −|an+1| )=cn−1 , en and, since c0 = αe1, (4.8) cn = αen+1. Besides, (4.6) and (4.7) yield βan cn−1an+1 + βan =0 ⇒ an+1 = − . cn−1 On the other hand, from (4.5) and (4.6) we get cn−1 + βanan+1 = cn, which implies 2 2 |β| |an| (4.9) cn−1 − = cn. cn−1 % % % %2 % β % 1 From (4.8) and (4.9), we find that en+1 + % % is independent of n.Thus, α en % % % % % %2 % %2 % β % 1 % β % β (4.10) en+1 + % % = e1 + % % =2 e , α en α α β where we have used (4.4) to express e =1−|a |2. Denoting γ = , (4.10) becomes 1 1 α 1 e + |γ|2 =2 e(γ), n+1 e n e γ which, with the notation x = n ,ω= e ,readsas n |γ| |γ| 1 (4.11) xn+1 + =2ω. xn This gives an expansion of xn as a continued fraction 1 | 1 | 1 | 1 | 1 xn =2ω − − −···− − ,x0 = , |2ω |2ω |2ω |x0 |γ| with solution x0Un(ω) − Un−1(ω) xn = , x0Un−1(ω) − Un−2(ω) 80 M. J. CANTERO, L. MORAL, AND L. VELAZQUEZ´ where Uk are the Chebyshev polynomials of second kind, given by λk+1 − λ−(k+1) γ U (ω)= ,λ= . k λ − λ−1 |γ| The parameter λ is one of the roots of the characteristic polynomial of (4.11), 2 z − 2ωz + 1. It is not important which root we choose to define Uk because they are inverse of each other. Then, the coefficients en are given by Un(ω; |γ|) (4.12) en = |γ| , Un−1(ω; |γ|) where Un(ω; |γ|)=Un(ω)−|γ|Un−1(ω) are the co-recursive Chebyshev polynomials (see [6]). The quasi-definite character of u is given by the condition en =0, ∀n ≥ 0. Then, the choice of the parameter a1, which is equivalent to the choice of β,mustbe such that Un(ω; |γ|), ∀n ≥ 0. In this case we can compute for the Schur parameters an, giving also an explicit expression for the orthogonal polynomials ϕn. ∗ − Starting with Bϕn = Cn−1ψn + Dn−1ψn and denoting ζ = β/β, from (4.6) and (4.8), n β(z − ζ)ϕn(z)=(βz + αen)z + βan. This equation is solved by − n βζ + αen n en β an = − ζ = 1 − β γ β and αe − β zn − ζn ϕ (z)=zn + n . n β z − ζ This rational modification of the Lebesgue functional v provides infinitely many quasi-definite functionals u. For instance, the solution u is quasi-definite whenever γ is imaginary. Then, ω =0andx2n =1/|γ|, x2n+1 = −|γ|. Hence, e2n =1, 2 n e2n+1 = −|γ| ,whichgivesan =(−β/β) aˆn witha ˆ2n =1+1/γ,ˆa2n+1 =1+γ. The Stieltjes function of u(αz + α) can be written as (αz + α)S(z) − αμ0 in terms of S(z), the Stieltjes function of u. Therefore, the rational modification u(αz + α)=v(βz + β) yields a linear spectral transformation v → u characterized by the relation (βz + β)T (z)+αμ − βν S(z)= 0 0 , αz + α where μ0 = u[1], ν0 = v[1] and T (z) is the Stieltjes function of v. Since the functionals u, v are hermitian, the relation between their Carath´eodory functions F (z),G(z) comes from their connection with the Stieltjes functions, which gives (βz + β)G(z)+ν (βz − β) − μ (αz − α) F (z)= 0 0 . αz + α In the case of the Lebesgue functional v,theCarath´eodory function is G(z)= ν0 =1,thus 2βz − μ0(αz − α) w + z F (z)= = γ +(μ − γ)Δ− (z), Δ (z)= . αz + α 0 α/α w w − z SPECTRAL TRANSFORMATIONS OF HERMITIAN LINEAR FUNCTIONALS 81 Bearing in mind that Δw(z)istheCarath´eodory function of δ(z − w)andu must be hermitian, we conclude that $ α −1 γv +(μ − γ)δ(z + )inP∗ := C[z ], u = 0 α − α P γv +(μ0 γ)δ(z + α )in. 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Spain Current address:Mar´ıa de Luna,3 E-mail address: [email protected] Contemporary Mathematics Volume 578, 2012 http://dx.doi.org/10.1090/conm/578/11468 Numerical study of higher order analogues of the Tracy–Widom distribution T. Claeys and S. Olver Dedicated to Paco Marcell´an on the occasion of his 60th birthday Abstract. We study a family of distributions that arise in critical unitary random matrix ensembles. They are expressed as Fredholm determinants and describe the limiting distribution of the largest eigenvalue when the dimension of the random matrices tends to infinity. The family contains the Tracy– Widom distribution and higher order analogues of it. We compute the distri- butions numerically by solving a Riemann–Hilbert problem numerically, plot the distributions, and discuss several properties that they appear to exhibit. 1. Introduction Consider the space of Hermitian n × n matrices with a probability measure of the form 1 (1.1) e−n tr V (M)dM, Zn where dM is the Lebesgue measure defined by n dM = Mjj dRe MijdIm Mij, j=1 i 2010 Mathematics Subject Classification. Primary 33E17, 35Q15, 65R10. TC acknowledges support by the Belgian Interuniversity Attraction Pole P06/02 and by the ERC program FroM-PDE. c 2012 American Mathematical Society 83 84 T. CLAEYS AND S. OLVER and is characterized as the unique measure μV which minimizes the logarithmic energy [8] 1 (1.3) I (μ)= log dμ(x)dμ(y)+ V (x)dμ(x), V |x − y| among all probability measures μ on R. The equilibrium measure μV depends on ∪ the external field V and is supported on a finite union of intervals j=1[aj,bj ], with , aj,bj depending on V . It is absolutely continuous with a density ψV of the form [11] − − ∈∪ (1.4) ψV (x)= (bj x)(x aj ) h(x),xj=1[aj ,bj ], j=1 where h(x) is a real analytic function on R. Generically [16], h has no zeros on the support and in particular at the endpoints aj ,bj , so that the equilibrium density has square root vanishing at the endpoints. However, there exist singular external fields V which are such that h(bj)=0(orh(aj)=0).Ifh(bj) = 0, then necessarily an odd number of derivatives vanish: we have (2k−1) (2k) (1.5) h (bj )=···= h (bj )=0,h(bj ) =0 , for k ∈ N. The value k = 0 corresponds to the generic square-root behavior and 5/2 k = 1 to the first type of critical behavior where ψV (x) ∼ c(bj − x) as x bj . Example 1.1. The simplest critical case k = 1 occurs, e.g., for the critical quartic potential 1 4 1 8 (1.6) V (x)= x4 − x3 + x2 + x. 20 15 5 5 The limiting mean eigenvalue density is then supported on [−2, 2] and given by [6] 1 (1.7) ψ (x)= (x +2)1/2(2 − x)5/2,x∈ [−2, 2]. V 10π Example 1.2. The general situation k ∈ N can be realized for instance with the following family of polynomials, 2 k+1 1 (2k +2)! (−1) ( + ) − (1.8) V (x)=2· 2 2k+1 x +1, k − 1 (2k +1− )!( +1) 2 2k+2 =0 where (a)m = a·(a+1) ···(a+m−1) is the Pochhammer symbol. The corresponding limiting mean eigenvalue densities are given by (2k +2)! (1.9) ψ (x)=− x1/2(1 − x)2k+1/2,x∈ [0, 1]. k − 1 π 2 2k+2 This can be verified by substituting (1.9) into the variational conditions that char- acterize the equilibrium measures in external field Vk. These conditions reduce to the equation 1 ψk(s) ∈ 2 − ds = Vk(x), for x (0, 1), 0 x s which is readily verified. NUMERICAL STUDY OF HIGHER ORDER TRACY-WIDOM DISTRIBUTIONS 85 The random matrix ensembles under consideration are known to have eigen- values following a determinantal point process with a kernel of the form [8] − n V (x) − n V (y) e 2 e 2 κn−1 (1.10) Kn(x, y)= (pn(x)pn−1(y) − pn(y)pn−1(x)), x − y κn −nV (x) where pj is the degree j orthonormal polynomial with respect to the weight e on R; κj is its leading coefficient. Near a right endpoint b of the support for which k =0,itwasproved[12, 9] that the large n limit of the re-scaled kernel is the Airy kernel: 1 u v (0) (1.11) lim Kn b + ,b+ = K (u, v), n→∞ cn2/3 cn2/3 cn2/3 (0) uniformly for u, v ≥−L, L>0, where c = cV and K is the Airy kernel Ai (u)Ai (v) − Ai (v)Ai (u) (1.12) K(0)(u, v)= . u − v s The limit of the probability that the largest eigenvalue is smaller than b + cn2/3 is given by a Fredholm determinant: 2/3 − − (0) (1.13) lim Prob cn (λn b) (1.16) q0(x) ∼ Ai (x), as x → +∞, −x 1 (1.17) q (x)= 1+ + O x−6 , as x →−∞. 0 2 8x3 If b is a right endpoint of the support for which k = 1, i.e. (1.5) holds with k =1andbj = b, the kernel has a different limit. It was showed in [7], see also [4, 5], that there exists a limiting kernel K(1) such that 1 u v (1) (1.18) lim Kn b + ,b+ = K (u, v), n→∞ cn2/7 cn2/7 cn2/7 uniformly for u, v in compact subsets of the real line. Since both sides of this equation tend to 0 rapidly as u or v tends to +∞, this result can be extended to a uniform statement for u, v ≥−L, L>0, which implies 2/7 − − (1) (1.19) lim Prob cn (λn b) 86 T. CLAEYS AND S. OLVER consider an external field V = Vt depending on a parameter t, it can happen that the support consists of two intervals for t<1, but of only one interval for t =1. This happens for example when two intervals approach each other and simultane- ously one of the intervals shrinks in the limit t 1. Such phase transitions are (1) covered by the kernels K (u, v; t0,t1) with non-zero parameters t0,t1. We refer the reader to [7]and[6] for more information on this issue. (1) The kernels K (u, v; t0,t1) are related to the second member of the Painlev´e I hierarchy, but are most easily characterized in terms of a Riemann-Hilbert (RH) problem. We will give more details about this in the next section. Although no proofs have been given for k>1, when considering in more detail the results and methods in [7], one expects a result of the form (1.20) 1 u v (k) lim Kn b + ,b+ = K (u, v; t0,...,t2k−1), n→∞ cn2/(4k+3) cn2/(4k+3) cn2/(4k+3) for general k>1, where the kernel now depends on 2k parameters, and 2/(4k+3) − − (k) (1.21) lim Prob cn (λn b) 4k+3 − (k) −cs 2 → ∞ (1.22) log det(I Ks (t0,...,t2k−1)) = O(e ), as s + . The asymptotics as x →−∞are more subtle and require a detailed analysis of the Fredholm determinants. In the simplest case t0 = ··· = t2k−1 =0,theyaregiven by 1 Γ(2k + 3 )2 (1.23) log det(I − K(k))=− 2 |s|4k+3 s 3 2 2 4(4k +3)Γ( 2 ) Γ(2k +2) 2k +1 (k) − 4k+3 − log |s| + χ + O(|s| 2 ), as s →−∞, 8 where Γ(x) is Euler’s Γ-function. The constant χ(k) has no explicit expression, (0) 1 − except for k = 0, where it was proved in [10, 1]thatχ = 24 log 2 + ζ ( 1), and ζ(s) is the Riemann zeta function. The goal of this paper is to set up a numerical scheme for computing the Fred- (k) holm determinants det(I − Ks (t0,...,t2k−1)), which will allow us to draw plots of the distributions and their densities, to verify formulas (1.22) and (1.23) nu- merically, to compute numerical values for the constants χ(k), and to formulate a number of questions about the analytic properties of the distributions (monotonic- ity, inflection points), based on a closer inspection of the plots. In the next section, we define the kernels in a precise way using a RH problem. This RH characteriza- tion will also be used for the numerical analysis which we explain in more detail in (k) Section 3. In Section 4 finally, we show plots of the distributions det(I − Ks )and NUMERICAL STUDY OF HIGHER ORDER TRACY-WIDOM DISTRIBUTIONS 87 10 11HH H Γ2 HH HjH HH 01 Γ3 - Hr - Γ1 11 − ¨ 10 ¨¨ 0 01 *¨ ¨¨ ¨ ¨ Γ4 10¨¨ 11 Figure 1. The jump contour Γ and the jump matrices for Φ. their densities for several values of k and the parameters t0,...,t2k−1, and we will formulate a number of open problems. 2. Riemann–Hilbert characterization of the kernels The kernels K(k) have the form (2k) (2k) − (2k) (2k) (k) Φ1 (u)Φ2 (v) Φ1 (v)Φ2 (u) (2.1) K (u, v; t ,...,t − )= , 0 2k 1 −2πi(u − v) (2k) (2k) where the functions Φj (w)=Φj (w; t0,...,t2k−1) can be characterized in terms of the following RH problem; they are defined below the RH problem. RH problem for Φ. (a) Φ = Φ(2k) : C \ Γ → C2×2 is analytic, with − −iπ − iπ − ∪4 ∪{ } R+ R 4k+3 R 4k+3 R Γ= j=1Γj 0 , Γ1 = , Γ3 = , Γ2 = e , Γ4 = e , oriented as in Figure 1. (b) Φ has continuous boundary values Φ+ as ζ approaches Γ \{0} from the left, and Φ−, from the right. They are related by the jump conditions (2.2) Φ+(ζ)=Φ−(ζ)Sj, for ζ ∈ Γj, where 11 (2.3) S = , 1 01 10 (2.4) S = S = , 2 4 11 01 (2.5) S = . 3 −10 (c) Φ has the following behavior as ζ →∞: 1 − σ3 −1/2 −1 −θ(ζ)σ3 (2.6) Φ(ζ)=ζ 4 N I + hσ3ζ + O(ζ ) e , 88 T. CLAEYS AND S. OLVER where h = h(t0,...,t2k−1) is independent of ζ, σ3 is the Pauli matrix 10 , N is given by 0 −1 1 11− 1 πiσ (2.7) N = √ e 4 3 , 2 −11 and − 2 k 1 j 2 4k+3 (−1) tj 2j+1 (2.8) θ(ζ; t ,...,t − )= ζ 2 − 2 ζ 2 , 0 2k 1 4k +3 2j +1 j=0 where the fractional powers are the principal branches analytic for ζ ∈ C \ (−∞, 0] and positive for ζ>0. (d) Φ is bounded near 0. Itwasprovedin[6] that this RH problem is uniquely solvable for any real values (2k) (2k) of t0,...,t2k−1. The functions Φ1 =Φ1 and Φ2 =Φ2 appearing in (2.1) are the analytic extensions of the functions Φ11 and Φ21 from the sector in between Γ1 and Γ2 to the entire complex plane. Alternatively they can be characterized as fundamental solutions to the Lax pair associated to a special solution to the 2k-th member of the Painlev´e I hierarchy. We will not give details concerning this alternative description, since the RH characterization is more direct and more convenient for our purposes. Remark 2.1. The description in terms of differential equations in the PI hier- archy presents the possibility of computing these distributions using ODE solvers. However, similar to the Hastings–McLeod solution (see [21]), these solutions are inherently unstable as initial value problems; hence, applying initial value solvers requires the use of high precision arithmetic, which is too computationally expen- sive to be practical. On the other hand, the Hastings–McLeod solution can be solved reliably as a boundary value problem [3, 14]. In experiments conducted in the chebfun package [20], it appears that boundary value problem solvers can also be used for the higher order analogues. However, the extreme high order of the differential equation — it is order 4k + 2 — results in large inaccuracies in the computed solution. Furthermore, the differential equations become increasingly complicated as k increases, inhibiting the efficiency of the automatic differentiation required in the solver. We therefore do not take this approach further. On the other hand, the representation in terms of a RH problem is numerically stable, and therefore is reliable. Not only the kernel K(k) can be described in terms of a RH problem, but also the logarithmic derivative of the Fredholm determinant can be expressed in terms of a RH problem, which shows similarities with the above one, but is nevertheless genuinely different. We have a formula of the form % d (k) 1 −1 % (2.9) log det(I − K (t ,...,t − )) = X (ζ)X (ζ) , ds s 0 2k 1 2πi s s 21 ζs where Xs is the unique solution to a RH problem, see [6, Section 2]. This representation provides relative accuracy, whereas the representation as a Fredholm determinant only provides absolute accuracy [2]. However, it requires NUMERICAL STUDY OF HIGHER ORDER TRACY-WIDOM DISTRIBUTIONS 89 solving a RH problem for each point of evaluation s and numerical indefinite integra- tion to recover the distributions. Therefore, the expression in terms of a Fredholm determinant is more computationally efficient. 3. Numerical study of the distributions We will compute the higher order Tracy–Widom distributions by calculating Φ numerically, using the methodology of [18, 17]. Consider the following canonical form for a RH problem: Canonical form for RH problem for Ψ. (a) Ψ : C \ Γ → C2×2 is analytic, where Γ is an oriented contour which is the closure of the set Γ = Γ1 ∪···∪Γ whose connected components can be M¨obius-transformed to the unit interval Mi :Γi → (−1, 1), with junction points Γ∗ = Γ \ Γ. (b) Ψ has continuous boundary values Ψ+ as ζ approaches Γ from the left, and Ψ−, from the right. For a given function G, they are related by the jump condition (3.1) Ψ+(ζ)=Ψ−(ζ)G(ζ). (c) As ζ →∞, we have lim Ψ(ζ)=I. (d) Ψ is bounded near Γ∗. Define the Cauchy transform C 1 f(t) Γf(ζ)= − dt, 2πi Γ t ζ ∈ C+ C− and denote the limit from the left (right) for ζ Γby Γ ( Γ ). We represent Ψ in terms of the Cauchy transform of an unknown function U defined on Γ: Ψ(ζ)=I + CΓU(ζ). Plugging this into (3.1) we have the linear equation C+ −C− − (3.2) Γ U Γ UG = G I. We solve this equation using a collocation method. Γ1 Γ We approximate U by Un for n = {n ,...,n }, which is defined on each component Γi of the contour in terms of a mapped Chebyshev series: n Γi −1 Γi ∈ Un(x)= Uj Tj (Mi(x)), for x Γi and i =1,...,, j=0 Γi ∈ C2×2 where Uj ,andTj is the j-th Chebyshev polynomial of the first kind. The C ◦ convenience of this basis is that the Cauchy transforms Γi [Tj Mi]areknownin closed form, in terms of hypergeometric functions which can be readily computed numerically [19]. ∗ For each ζ ∈ Γ ,letΩ1,...,ΩL be the subset of components in Γ that have ζ as an endpoint. In other words, Mi(ζ)=pi where pi = ±1fori =1,...,L.We say that U satisfies the zero sum condition if L Ωi piU (ζ)=0, i=1 90 T. CLAEYS AND S. OLVER Ωi where U denotes U restricted to Ωi. The boundedness of Ψ implies that U must satisfy the zero sum condition. Define the mapped Chebyshev points of the first kind: ⎛ ⎞ −1 ⎜ ⎟ ⎜cos π 1 − 1 ⎟ ⎜ nΓi −1 ⎟ Γ −1 ⎜ ⎟ x i = M ⎜ . ⎟ i ⎜ . ⎟ ⎝ cos π ⎠ nΓi −1 1 and the vector of unknown Chebyshev coefficients (in C2×2) ⎛ ⎞ Γ1 U0 ⎜ . ⎟ U = ⎝ . ⎠ . U Γ nΓ −1 Then we can explicitly construct a matrix C− such that ⎛ − ⎞ C Γ1 Γ Un(x ) − ⎜ . ⎟ C U = ⎝ . ⎠ − C Γ Γ Un(x ) − C Γi holds whenever Un satisfies the zero sum condition [17]. To define Γ Un(x )at the endpoints, we use − −1 − −1 C Un(M (±1)) = lim C Un(M (x)), Γ i x→±1 Γ i which exists when Un satisfies the zero sum condition. Thus we discretize (3.2) by − − LnU =(I + C )U − C UGn = Gn − I Γ Γ where Gn =(G(x 1 ),...,G(x )) and the multiplication by Gn on the right is defined in the obvious way. The remarkable fact is that solving this linear system will generically imply that Un satisfies the zero sum condition if Ln is nonsingular; if it does not, Ln is necessarily not of full rank, and we can replace redundant rows with conditions imposing the zero sum condition [17]. Taking this possibly modified definition of Ln, we have the following convergence result. Theorem 3.1. [17] The L∞ error of the numerical method is bounded by −1 | − ¯ | Cn Ln ∞ U Un , where Cn grows logarithmically with max n, U¯n is the polynomial which interpolates U at xΓ1 ,...,xΓ and | | −1 f = f ∞ +max (M ) fi ∞. i i −1 In practice, Ln appears to grow at most logarithmically with max n when- ever a solution to the RH problem exists. Therefore, if the solution U is smooth, the numerical method will converge spectrally as min n →∞, with min n proportional to max n. To apply the numerical method to the RH problem Φ, we need to reduce it to −σ /4 −θ(ζ)σ canonical form. Define W (ζ)=ζ 3 Ne 3 , and we use the notation W± to NUMERICAL STUDY OF HIGHER ORDER TRACY-WIDOM DISTRIBUTIONS 91 denote the analytic continuation of W above/below its branch cut along (−∞, 0). We make the following transformation: ⎧ ⎪W (ζ) |ζ| > 1 ⎪ ⎪ −1 | | ⎨S4 ζ < 1andζ lies between Γ4 and Γ1 (2k) (2k) −1 | | (3.3) Φ (ζ)=Ψ (ζ) ⎪S4 S1 ζ < 1andζ lies between Γ1 and Γ2 . ⎪ −1 −1 ⎪S S1S |ζ| < 1andζ lies between Γ2 and Γ3 ⎩⎪ 4 2 I |ζ| < 1andζ lies between Γ3 and Γ4 Then Ψ(2k) satisfies the RH problem: RH problem for Ψ(2k). (a)Ψ=Ψ(2k) : C \ Γ˜ → C2×2 is analytic, with −iπ iπ Γ=˜ Γ˜1 ∪ Γ˜2 ∪ Γ˜4 ∪ Γ˜21 ∪ Γ˜42 ∪ Γ˜14 ∪{1,e4k+3 ,e4k+3 }, −iπ iπ Γ˜1 =(1, ∞), Γ˜2 = −e 4k+3 (∞, 1), Γ˜4 = −e 4k+3 (∞, 1), iπ(1− 1 ,0) iπ(−1+ 1 ,1− 1 ) iπ(0,−1+ 1 ) Γ˜21 = e 4k+3 , Γ˜42 = e 4k+3 4k+3 , Γ˜14 = e 4k+3 oriented as in Figure 2. (b) The jump conditions for Ψ are given by −1 Ψ+(ζ)=Ψ−(ζ)W (ζ)SjW (ζ), for ζ ∈ Γ˜j , j =1, 2, 4, −1 −1 ∈ ˜ Ψ+(ζ)=Ψ−(ζ)S4 W (ζ), for ζ Γ14, −1 −1 ∈ ˜ Ψ+(ζ)=Ψ−(ζ)S4 S1W (ζ), for ζ Γ21, −1 Ψ+(ζ)=Ψ−(ζ)W− (ζ), for ζ ∈ Γ˜42. (c) As ζ →∞, we have lim Ψ(ζ)=I. −iπ iπ (d) Ψ is bounded near {1,e4k+3 ,e4k+3 }. With Ψ in this form, we can readily compute it numerically, recover Φ by (k) (3.3), and thence evaluate the kernel of Ks numerically. This leaves one more task: computing the Fredholm determinant itself. We accomplish this using the framework of [2], which also achieves spectral accuracy. 4. Plots and open problems 4.1. Local maxima of the densities. In Figure 3, we plot the numerically computed distributions Fk(s;0,...,0) for k =0,...,5, where we write − (k) (4.1) Fk(s; t0,...,t2k−1)=det(I Ks (t0,...,t2k−1)). In Figure 4 the corresponding densities are drawn. One observes that each of the densities has only one local maximum (i.e. the distributions have only one inflection point). The figures suggest that for any k ∈ N and for t0 = ... = t2k−1 =0,the densities have only one local maximum. For general values of the parameters t0,...,t2k−1 ∈ R, the situation is different. WeseeinFigure5,fork =1,t0 = 0 and varying negative t1, that the densities have two local maxima. From the random matrix point of view, this can be explained (1) heuristically by the fact that the kernels K (u, v;0,t1)fort1 < 0 correspond to a double scaling limit which describes the transition from a random matrix model with a two-cut support (for the limiting mean eigenvalue distribution) to a one- cut support, where the parameter t1 regulates the speed of the transition. To be 92 T. CLAEYS AND S. OLVER HH HH H ˜ HΓ2 Hj H ˜ HH '$-Γ21 HH - Γ˜1 Γ˜42 ¨ ¨¨ &% ¨ ˜ *¨ Γ14 ¨¨ ¨ Γ˜ ¨¨ 4 ¨¨ Figure 2. The jump contour Γ˜ and the jump matrices for Ψ. 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 2 4 2 4 Figure 3. The distributions Fk for k =0, 1,...,5 with tj =0. The slope steepens near −1whenk increases. On the right, we also plot F∞ (thick curve), as constructed in Section 4.4. more precise, consider a random matrix ensemble with probability measure (1.1), where V = Vn depends on n. If the dependence of V on n is fine-tuned in an appropriate way, it can happen that the equilibrium measure μVn consists of two intervals for finite n, but of only one interval in the limit n →∞.Inorderto (1) obtain K (u, v;0,t1) as a scaling limit of the eigenvalue correlation kernel, both intervals in the support of μVn should approach each other and simultaneously one of the intervals should shrink, as n →∞.Ifthen-dependence of V is chosen in an appropriate way, the limiting probability that a random matrix has an eigenvalue located in the shrinking interval lies strictly between 0 and 1 (it actually increases when t1 decreases). We believe that one local maximum of the densities in Figure 5 (the one most to the left) corresponds to the largest eigenvalue if no eigenvalues lie in the shrinking interval, and the second local maximum corresponds to the largest NUMERICAL STUDY OF HIGHER ORDER TRACY-WIDOM DISTRIBUTIONS 93 4 3 2 1 2 4 Figure 4. The densities Fk(s)fork =0, 1,...,5 with tj =0. 1.0 1.5 0.8 0.6 1.0 0.4 0.5 0.2 1 2 3 1 2 3 Figure 5. The distribution F1(s) (left) and density F1(s)(right) for t0 =0andt1 = −1,...,−4. eigenvalue if this one lies in the shrinking interval. For k ∈ N, transitions can take place from at most k + 1 cuts to a one-cut regime, and for that reason we expect that for k ∈ N, the density function has at most k + 1 local maxima, although we have no analytical evidence for this. 4.2. Asymptotics as x → +∞. In Figure 6 we show the rate of convergence to one as s →∞of Fk(s) for various values of k.Fors<1, we see that the distri- bution appears to approach a fixed distribution. For s>1, the rate of convergence becomes increasingly rapid, matching the asymptotic formula (1.22). 94 T. CLAEYS AND S. OLVER 10 10 10 10 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Figure 6. 1 − Fk(s)fork =1,...,5 with tj =0. 4.3. Asymptotics as x →−∞. We note that the constants χ(k) in (1.23) can be expressed as χ(k) = lim log det(I − K(k)) − A(k) s→−∞ s s M = lim log det(I − K(k)) − ∂ log det(I − K(k))ds − A(k) , →−∞ M s s s s s with 1 Γ(2k + 3 )2 2k +1 A(k) = − 2 |s|4k+3 − log |s|. s 3 2 2 4(4k +3)Γ( 2 ) Γ(2k +2) 8 − − (k) For moderate M (we use M = .5), we can reliably calculate log det(I KM ) (k) as before. For s NUMERICAL STUDY OF HIGHER ORDER TRACY-WIDOM DISTRIBUTIONS 95 Error in approximating , scaled by |s|3 0 0.10 Errorin approximating 0 0.1 0.08 0.001 0.06 10 0.04 10 0.02 250 200 150 100 50 0 200 150 100 50 0 (0) Figure 7. The absolute error in approximating χ0, |χ − (0) (0) log det(I − Ks )+As | as a function of s (left). The error multi- plied by |s|3 (right), showing faster convergence than predicted. Using this approach, we estimate the first three χ(k): χ(0) ≈−0.1365400105, (matches exact expression to 8 digits) χ(1) ≈−0.09614954 and χ(2) ≈−0.06145. Cancellation and other numerical issues cause the approach to be unreliable for larger k. The convergence to χ(0) is verified in Figure 7. One interesting thing to note is that the rate of convergence appears to be faster than predicted: numerical evidence suggest convergence like O(|s|−3). A similar experiment for χ(1) suggests a convergence rate of O(|s|−7). (The numerics for χ(2) are insufficiently accurate to make a prediction.) Therefore, we conjecture that the error term in (1.23) is in −(4k+3) − 4k+3 fact O(|s| ), which is better than the theoretical error O(|s| 2 ). 4.4. Large k limit. For increasing k, one observes from Figure 3 that the slope of the distributions near −1 gets steeper. At first sight, one may expect from Figure 3 that for large k, the distribution function tends to a step function, but a closer inspection reveals that this is not the case. Instead, we conjecture that there is a limit distribution supported on [−1, 1] which is possibly discontinuous at −1 but continuous at 1. We present an asymptotic–numerical argument that this is indeed true. Con- (2k) −1 sider the RH problem for Ψ .NotethatonΓ˜j ,thejumpsWSjW → I as ∞ − k →∞. Furthermore, inside the unit circle W (ζ) → W ( )(ζ)=ζ σ3/4N. Finally, Γ˜42 disappears. Thus, in a formal sense, we have the following RH problem: RH problem for Ψ(∞). (a)Ψ=Ψ(∞) : C \ Γ → C2×2 is analytic, with iπ(1,0) iπ(0,−1) Γ=Γ˜21 ∪ Γ˜14 ∪{±1}, Γ˜21 = e , Γ˜14 = e . (b) The jump conditions for Ψ are given by (for W = W (∞)) −1 −1 ∈ ˜ Ψ+(ζ)=Ψ−(ζ)S4 W (ζ), for ζ Γ14, −1 −1 ∈ ˜ Ψ+(ζ)=Ψ−(ζ)S4 S1W (ζ), for ζ Γ21. 96 T. CLAEYS AND S. OLVER (c) As ζ →∞, we have lim Ψ(ζ)=I. This is not in canonical form: the jump matrices are not continuous at ±1, implying that the solution Ψ(∞) has singularities. We rectify this by using local parametrices to remove the jumps. Define ⎧ ⎪ 1 − z+i − ⎪ 10 1 log i − 1 10 ⎪ 2πi z i , |ζ − 1| NUMERICAL STUDY OF HIGHER ORDER TRACY-WIDOM DISTRIBUTIONS 97 operator acts only on L2(s, 1), not L2(s, ∞). Instead, we justify this by the accuracy of the numerics.) It should be noted that the local parametrices P (±1) are not close to the identity matrix on |ζ ∓ 1| = r, and therefore they would not be suitable parametrices to be used for a rigorous Deift/Zhou steepest descent analysis [13] applied to the RH problem for Ψ. However, this is not an issue here: numerically it is sufficient that the local parametrices satisfy the required jump conditions. While this construction has not been mathematically justified, it is perfectly usable in a numerical way. 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Spohn, Exact scaling functions for one-dimensional stationary KPZ growth, J. Stat. Phys. 115 (2004), 255–279. MR2070096 (2005d:82097) [22] C.A. Tracy and H. Widom, Level spacing distributions and the Airy kernel, Comm. Math. Phys. 159 (1994), no. 1, 151–174. MR1257246 (95e:82003) Universite´ Catholique de Louvain, Institut de Recherche en Mathematique´ et physique, Chemin du Cyclotron 2, B-1348 Louvain-La-Neuve, Belgium E-mail address: [email protected] School of Mathematics and Statistics, The University of Sydney, NSW 2006 Aus- tralia E-mail address: [email protected] Contemporary Mathematics Volume 578, 2012 http://dx.doi.org/10.1090/conm/578/11472 Comb functions Alexandre Eremenko and Peter Yuditskii Dedicated to the memory of Franz Peherstorfer Abstract. We discuss a class of regions and conformal mappings which are useful in several problems of approximation theory, harmonic analysis and spectral theory. 1. Introduction We begin with two classical problems which serve as motivation. Then in sections 2–5 we describe some classes of regions, corresponding conformal maps and entire and subharmonic functions. In sections 6–7 we discuss various problems where these classes appear. 1. Polynomials of least deviation from zero. Let E ⊂ R be a compact set on the real line, and Pn a polynomial with minimal sup-norm Ln = Pn E among all monic polynomials of degree n. If n 1991 Mathematics Subject Classification. Primary 30C20, 41A10, 47B36, 41A50. Key words and phrases. Conformal map, Green function, Martin function, uniform approx- imation, Jacobi matrices, Riesz bases, spectral theory. The first author was supported by NSF grant DMS-1067886. The second author was supported by the Austrian Science Fund FWF, project no: P22025- N18. c 2012 American Mathematical Society 99 100 ALEXANDRE EREMENKO AND PETER YUDITSKII Figure 1. Comb regions of V (left) and MO (right) types. upper half-plane, which is real on the real line. So it extends to an entire function, and the behavior at ∞ shows that this entire function is a polynomial of degree n.ChooseL so that the polynomial P = L cos θ is monic. It is easy to check that our extremal polynomials satisfying (i)–(iii) are of this form, with an appropriate ≥ choice of parameters hj 0. The set E is contained in E := θ−1([πm, πk]). This set E is the maximal extension of E, for which the extremal polynomial is the same as the one for E. Critical points of Pn are preimages of the tips of the slits under θ, and critical ± values are cosh hj. The θ-preimages of the points πj, m < j < k are solutions of Pn(z)=±L, and all these solutions are real. − For example, if E =[ 1, 1] we take all hj =0,andPn is the n-th Chebyshev polynomial. If E consists of two intervals symmetric with respect to 0, and n is even, we take all hk = 0, except one, h(m+k)/2 > 0. On polynomials of least deviation from 0 on several intervals we refer to [2, 3], [4, vol. 1] and the survey [47], where the representation Pn = L cos θ is used systematically. 2. Spectra of periodic Jacobi matrices. Consider a doubly infinite, periodic Jacobi matrix ⎛ ⎞ ...... ⎜ ⎟ ⎜ p−1 q−1 p0 000⎟ ⎜ ⎟ ⎜ 0 p0 q0 p1 00⎟ J = ⎜ ⎟ ⎜ 00p1 q1 p2 0 ⎟ ⎝ ⎠ 000p2 q2 p3 ...... which is constructed of two periodic sequences of period n,whereqj are real, and pj > 0. This matrix defines a bounded self-adjoint operator on 2,andwewishto describe its spectrum [18, 33, 45]. To do this we consider a generalized eigenvector u ∈ ∞ which satisfies Ju = zu, z ∈ C. COMB FUNCTIONS 101 For fixed z, this can be rewritten as a recurrent relation on the coordinates of u: pj+1uj +(qj+1 − z)uj+1 + pj+2uj+2 =0, which we rewrite in the matrix form as u 01/p u j+1 = j+1 j . pj+2uj+2 −pj+1 (z − qj+1)/pj+1 pj+1uj+1 Thus un u0 = Tn(z) , pn+1un+1 p1u1 where Tn(z) is a polynomial matrix with determinant 1, which is called the transfer- matrix. To have a bounded generalized eigenvector u, both eigenvalues of Tn must have absolute value 1. This happens if and only if |Pn(z)| := | tr Tn(z)|/2 ≤ 1. As Pn is a real polynomial, the spectrum is the preimage of the interval [−1, 1]. As our matrix J is symmetric, the spectrum must be real, this is the same as the condition that all solution of the equations Pn(z)=±1 are real, so we obtain a polynomial of the same kind as in Example 1. For every real polynomial with this −1 − property, there exists a periodic Jacobi matrix whose spectrum is Pn ([ 1, 1]), and all matrices J with a given spectrum can be explicitly described [34, 45]. Our polynomial has a representation Pn =cosθ,whereθ is a conformal map of the upper half-plane onto a comb region D as in Example 1. We obtain the result that the spectrum of a periodic Jacobi matrix consists of the intervals – preimage of the real line under a conformal map θ. ≤ ≤ − We can prescribe an arbitrary sequence hj , 1 j n 1, construct a conformal map θ : H → D,whereD is the region shown in Fig. 1 (right), and the polynomial − j ± P =cosθ will have critical values ( 1) cosh hj and all solutions of P (z)= 1 will be real. Such polynomial P is defined by its critical values of alternating sign up to a change of the independent variable z → az + b, a > 0,b∈ R. Later we will show that any real polynomial with arbitrary real critical points is defined by its critical values up to a change of the independent variable z → az + b, a > 0,b∈ R. We conclude this introduction with several historical remarks. After the works of Sergei Bernstein, in the 1950s and 1960s the development of the Chebyshev approximation was clearly dominated by the study of entire functions of least devi- ation from zero, see, for example [1, p. 320-363]. In particular, Boas and Schaeffer [9](seealso[36]) proved under very general assumptions that the extremal function can be expressed in terms of the hyperelliptic integral $ ) p(z) f(z)=L sin dz . q(z) The new wave of interest in the comb polynomials is most likely related to the theory of integrable systems, see e.g. [7], and the application of the iteration theory to the spectral theory of almost periodic operators with Cantor type spectrum, see e.g. [19]. Franz Peherstorfer explicitly formulated both properties of the comb polynomials (as polynomials of the least deviation from zero and in connection with periodic Jacobi matrices) in [40], see also [41, 42, 47]. An attractive aspect of this circle of questions is the numerous connections with diverse classical problems in analysis: investigations of Abel on expressing elliptic 102 ALEXANDRE EREMENKO AND PETER YUDITSKII and hyperelliptic integrals in elementary functions; continued fractions and Pell’s equations; factorization of functions on Riemann surfaces; subharmonic majorants; and so on. For modern surveys of these questions we refer to [47, 31]. For other applications of comb functions we mention [10, 11]. In parallel with our paper Injo Hur and Christian Remling presented a special paper [20] on applications of comb functions in spectral theory of ergodic Jacobi matrices and we refer the reader to it in particular, for further references related to this important direction. 2. Comb representation of LP entire functions In both examples in the Introduction, the class of real polynomials P such that all solutions of P (z)=±1 are real appears. Evidently, all zeros of such polynomials must be real and simple. Here we discuss a representation of polynomials with real zeros, not necessarily simple, using conformal mappings and generalization of this representation to a class of entire functions. Let P be a non-constant real polynomial of degree n with all zeros real. Let ϕ =logP be a branch of the logarithm in the upper half-plane H.Then P n 1 −ϕ = − = − P z − z n=j j is an analytic function in H with positive imaginary part. Lemma 2.1. An analytic function ψ in H whose derivative has positive imagi- nary part is univalent. Proof. Suppose that ψ(z1)=ψ(z2), zj ∈ H,z1 = z2. Then − 1 ψ(z1) ψ(z2) − 0= − = ψ (z2 + t(z1 z2))dt, z1 z2 0 but the last integral has positive imaginary part and thus cannot be 0. It is easy to describe the image ϕ(H). By Rolle’s theorem, all zeros of P are real and we arrange them in a sequence x1 ≤ ...≤ xn−1 where each zero is repeated according to its multiplicity. Let cj = P (xj)bethecritical sequence of P .Then the region D = ϕ(H) is obtained from a strip by removing n − 1rays: 3 (2.1) D = {x + iy : πm Here m − k = n,andhj =log|ck−j|≥−∞. Thus (2.2) P =expϕ, where ϕ is a conformal map of the upper half-plane onto a region D we just described (see Fig. 1, left). Such regions will be called polynomial V -combs. Letter V in this notation is used because this representation was introduced by E. B. Vinberg in [49]. Now suppose that an arbitrary finite sequence hj ∈ [−∞, ∞), m COMB FUNCTIONS 103 We obtain Theorem 2.2. For every finite sequence c1,...,cn−1 with the property (2.3) cj+1cj ≤ 0, there exists a real polynomial with real zeros for which this sequence is the critical sequence. Such polynomial is defined by its critical sequence up to a real affine change z → az + b, a > 0 of the independent variable. Now we extend this result to entire functions. Recall that an entire function belongs to the class LP (Laguerre-P´olya) if it is a limit of real polynomials with all zeros real. For more information on the LP -class and its applications we refer to [25]. Consider the following class of regions. Begin with {x + iy : πm (2.4) {x + iπj : x ≤ hj},m An important subclass of LP is defined by the condition that hj ≥ 0, m 104 ALEXANDRE EREMENKO AND PETER YUDITSKII or from both sides), then the corresponding asymptotic value is ∞. This subclass of LP will be called the MO-class. It was used for the first time in spectral theory in [35]. Functions of MO-class have another representation in terms of conformal mappings. Consider a region D of the form k3−1 { }\ { ≤ ≤ } (2.5) x + iy : y>0,πm 3. MacLane’s theorem In this section we give a geometric characterization of integrals of LP func- tions. Roughly speaking, we will show that critical values of these integrals can be arbitrarily prescribed, subject to the evident restriction (3.1). Notice that dif- ferentiation maps LP into itself, so the class of integrals of LP -functions contains LP . We follow the exposition in [49] with some corrections and simplifications, see also [14] on related questions. Let f be a real entire function with all critical points real. Consider the preimage f −1(R). It contains the real line, and it is a smooth curve in a neighborhood of any point which is not a critical point At a critical point of order n it looks like the preimage of the real line under zn+1. MacLane’s class consists of real entire functions for which the preimage of the real line looks like one of the pictures in Fig. 2, up to an orientation preserving homeomorphism of the plane, commuting with the complex conjugation. We call this picture a fish-bone. There are several cases. In the simplest case, the sequence of critical points ... ≤ xj ≤ xj+1 ≤ ... is unbounded from below and from above. Each critical point is repeated in this sequence according to its multiplicity. Preimage of the real line consists of the real line itself, crossed by infinitely many simple curves, each curve is symmetric with respect to the real line. The crossing points are mapped onto the critical values cj = f(xj). Several “vertical” lines cross the real line at a multiple critical point. The complement to the union of curves in Fig. 2, consists of simply connected regions, each of them is mapped conformally onto the upper or lower half-plane. COMB FUNCTIONS 105 Figure 2. Fish-bones. The sequence of critical points can be bounded from above or from below or both. Suppose that it is bounded from below, and enumerate the sequence as x1 ≤ x2 ≤ .... Then the left end of the fish-bone can be of two types. For the first type, shown in Fig 2 (left), the full preimage of the real line is connected. We have two large complementary regions adjacent along a negative ray, each of them is mapped by f homeomorphically onto the upper or lower half-plane. It is easy to see that in this case we have f(x) →∞as x →−∞.Wesetc0 = ∞ and extend our critical sequence (cj ) by adding this term to it. The second type of the end is shown in Fig. 2 (right). In this case, the preimage of the real must have infinitely many components. In addition to one component of f −1(R), as above, there are infinitely many simple curves tending to infinity at both ends. Strip-like regions between these curves are mapped homeomorphically onto the upper or lower half-plane. In this case c0 = limx→−∞ f(x) = ∞,andwe extend our critical sequence by c0. Similar situations may occur on the right end when the sequence of critical points is bounded from above. In all cases, the fish-bone is completely determined by the augmented critical sequence (cj ). We use the following notation: if the sequence of critical points is unbounded from above and below, then −∞ (3.1) (cj+1 − cj )(cj − cj−1) ≤ 0. All cj are real, except possibly the first and/or the last term which can be ±∞. We call such sequences “up-down sequences”. If the sequence of critical points is unbounded from below and from above, then the sequences xj and cj are defined for a given f up to a shift of the subscript. MacLane’s theorem [30] For every up-down sequence, finite or infinite in one or both directions, there exists a function f ∈ M for which this sequence is the critical sequence. Any two functions corresponding to the same sequence are related by f1(z)=f2(az + b) with a>0,b∈ R. 106 ALEXANDRE EREMENKO AND PETER YUDITSKII In other words, one can prescribe a piecewise-monotone graph on the real line, and after a strictly increasing continuous change of the independent variable, this will be the graph of an entire function of MacLane’s class, which is essentially unique. Uniqueness statement in MacLane’s theorem is easy. Suppose that f1 and f2 are two functions of MacLane class with the same augmented critical sequence. Then it is easy to construct a homeomorphism φ of the plane such that f1 = f2 ◦ φ. Then φ must be conformal and commute with complex conjugation, so φ must be arealaffinemap. Class LP is contained in the MacLane class. It corresponds to the case when the critical sequence satisfies the condition (2.3) and in addition, the first and last terms of the sequence cj ,ifpresent,are0or∞. It is clear that (2.3) is stronger than (3.1). We proved this special case of MacLane’s theorem in the previous section. Now we give the proof of MacLane’s theorem in full generality. First we recover the fish-bone from the given sequence (cj) as explained above. Then we construct a continuous map F : Cz → Cw as follows. We map each interval [xj ,xj+1] ∈ R linearly onto the interval [cj ,cj+1]. Then we map each infinite ray of the fish-bone onto a corresponding ray of the real line, linearly with respect to length. The curves on the left of Fig. 2 (right) are mapped on the rays [c0, ∞). Then we extend our map to the components of the complement of the fish-bone, so that each component is mapped on the upper or lower half-plane homeomorphically. The resulting continuous map F is a local homeomorphism everywhere except the points xj where it is ramified. There is unique conformal structure ρ in the plane Cz which makes this map holomorphic. By the uniformization theorem, the simply connected Riemann surface (C,ρ) is conformally equivalent to a disc |z| Then fn maps univalently some disc {z : |z| COMB FUNCTIONS 107 Figure 3. Approximation of a fish-bone by polynomial ones. Now we use the following lemma [25]. Lemma 3.1. Let gn be a sequence of real polynomials whose all zeros are real, and suppose that gn → g ≡ 0 uniformly in some neighborhood of 0.Theng is entire, and gn → g uniformly on compact subsets of C. Proof. By a shift of the independent variable we may assume that g(0) =0 . Then gn(0) = 0 for large n.Wehave − gn 1 (0) = 2 , gn z k n,k where zn,k are zeros of gn. The left hand side is bounded by a constant independent of n, while all summands in the right hand side are positive. So for every interval I on the real line there exists a constant c(I) independent of n such that the gn have at most c(I)rootsonI. Thus from every sequence of gn one can choose a subsequence such that the zero-sets of polynomials of this subsequence tend to a limit set which has no accumulation points in C. So our subsequence converges to an entire function. Evidently this entire function is an analytic continuation of g, and the statement of the lemma follows. We apply this lemma to the sequence (fn) and conclude that f is entire, that is R = ∞, as advertised. Now we describe the necessary modifications of this proof for the case that the sequence of critical points is bounded from below (the case of semi-infinite sequence bounded from above is treated similarly). If the asymptotic value c0 = ∞,no modification is needed. If c0 = ∞, we may assume without loss of generality that c0 = 0, by adding a real constant to all functions f,F,fn. Then we approximate our critical sequence c0,c1 ... by the finite sequences c0,c0,...,c0,c1 ...,cn,where c0 =0isrepeatedn times. The corresponding fish-bone is shown in Fig. 3, where β is the additional zero of multiplicity n.Asn →∞, β →−∞. The rest of the argument goes without change. Finally we consider the case when there are finitely many critical points and two different# asymptotic values. In this case, no approximation argument is needed, z − 2 and f(z)= −∞ P (ζ)exp( aζ +bζ)dζ,whereP is a real polynomial with all zeros real and a ≥ 0andb ∈ R. 108 ALEXANDRE EREMENKO AND PETER YUDITSKII 4. Representation of Green’s and Martin’s functions Here we discuss the relation between comb regions and Green and Martin func- tions of complements of closed sets on the real line. Let E ⊂ R be a compact set of positive capacity. Then there exists the Green function G of Ω = C\E with pole at ∞.Wehave (4.1) G(z)= log |z − t|dμ(t)+γ(E), E where μ is a probability measure on E which is called the equilibrium measure, and γ the Robin constant of E. Function G is positive and harmonic in C\E, and has boundary values 0 a. e. with respect to μ.Wehave (4.2) G(z) = log |z| + γ + o(1),z→∞. These properties characterize G and μ [24]. There exists an analytic function φ : H → H, such that G =Imφ. It is called the complex Green function. Since the derivative d − dμ(t) φ = i log(z t)dμ(t) = i − dz E E z t has positive real part in H, we conclude from Lemma 2.1 that φ is univalent. Let D = φ(H). This region D has the following characteristic properties: (i) D is contained in a vertical half-strip {x + iy : a (ii) For every z ∈ D, the vertical ray z = {z + it : t ≥ 0} is contained in D. (iii) For almost every x ∈ (a, b), the ray {x + iy : y>0} is contained in D. These properties can be restated shortly as follows: (4.3) D = {x + iy : a COMB FUNCTIONS 109 can never hit the real line, and an analytic continuation of φ−1 is possible along the whole ray z. To prove (iii), we use (4.4) again. As μ(t) is continuous, Re φ is continuous in H.Moreover, Re φ(β) − Re φ(α)=μ(β) − μ(α),α<β. Thismeansthatmeasureμ on E corresponds to the Lebesgue measure on base of the comb (a, b). Furthermore, if for some x ∈ E we have G(x)=0thenh(Re φ(x)) = 0. Thus h = 0 almost everywhere with respect to the Lebesgue measure on (a, b). This proves (iii), NowweshowthatforeveryD satisfying (i)–(iii), the conformal map φ : H → D is related with the Green function G of some closed set E by the formula G =Imφ. Imaginary part v =Imφ is a positive harmonic function in the upper half- plane. We extend it to the lower half-plane by symmetry, v(z)=v(z), and to the real line by upper semicontinuity: v(x) = lim supz→x v(z). In view of (i), ∂D has a rectilinear part near infinity, the extended function v is harmonic in a punctured neighborhood of ∞ and has asymptotics of the form v(z) = log |z| +const+o(1),z→∞. Let us prove that v is subharmonic in the whole plane, and has a representation (4.1) with some probability measure μ with compact support on the real line. Let {hk} be a dense set on ∂D.LetDn be the region obtained from the half- strip {x+iy : a Im φn(z)= log |z − t|dμn(t)+γn, with some probability measures μn on [0, 1] and some constants γn.Wecanchoose a subsequence such that μn → μ weakly, where μ is a probability measure on [0, 1], and it is easy to check that (4.1) holds with some γ.Thusv is subharmonic in the plane. Since v ≥ 0, the measure μ has no atoms. It remains to prove that v(x)=0 a. e. with respect to μ. This follows from the property (iii) of the region D. Indeed, let x ∈ (a, b) be a point such that the vertical ray x is in D, except the endpoint x. By a well-known argument, the curve φ(x) has an endpoint at some x ∈ (0, 1), and the angular limit of v =Imφ is zero at this point x. By the remark above, v(x) = 0. We define E as the closed support of μ.Thenv =Imφ is positive and harmonic outside E and v(x)=0μ-almost everywhere, so v is the Green function of E. Our construction of D from E defines D up to a shift by a real number. The inverse construction defines E up to a real affine transformation, and changing E by a set of zero capacity. Now we give a similar representation of Martin functions. Let E ⊂ R be an unbounded closed set of positive capacity. Let U betheconeofpositiveharmonic functions in C\E,andUs ⊂ U the cone of symmetric positive harmonic functions, v(z)=v(z). Martin’s functions are minimal elements of U, that is functions v ∈ U with the property u ∈ U, u ≤ v implies u = cv, 110 ALEXANDRE EREMENKO AND PETER YUDITSKII where c>0 is a constant. Similarly we define symmetric Martin functions using Us instead of U. Martin functions always exist and form a convex cone. If v is a Martin function, then v(z)+v(z) is a symmetric Martin function, so symmetric Martin functions also exist and form a convex cone. Let v be a symmetric Martin function, and let φ be an analytic function in H so that v(z)=Imφ(z),z∈ H.Thenφ : H → D is a conformal map onto a region D ⊂ H. This is proved in the same way as for Green’s functions. Regions D arising from symmetric Martin functions are characterized by the properties (ii), (iii) above and the negation of the property (i): either a = −∞ or b =+∞,orh is unbounded in (4.3). Levin introduced the following classification of regions D: Class A: a = −∞ and b = ∞. Class B: one of the numbers a, b is finite, another infinite. Class C: both a and b are finite. Kesarev [21] gave a geometric criterion in terms of the set E which distinguish the cases A, B and C. Notice that function φ maps h into H, so the angular derivative of φ at infinity exists, that is φ(z)=cz + o(z),z→∞ in any Stolz angle, where c ≥ 0. One can derive from this that every Martin function satisfies B(r, v):=maxv(z)=O(r),r→∞. |z|=r This implies that the cone of Martin functions has dimension at most 2, [22, 29, 12], and the cone of symmetric Martin functions is always one-dimensional. Dimension of the cone of Martin’s functions is an important characteristic of the set E,see[8, 29]. One can show that the cone of Martin functions is two- dimensional if and only if lim sup B(r, v)/r > 0. r→∞ A geometric criterion which tells in terms of the set E the dimension of the cone of Martin’s functions is given by Andrievskii [6]. Now we impose various conditions on D and find their exact counterparts in terms of E and μ. The first important condition is that the set E is regular in the sense of potential theory [24]. In this case Green’s and Martin’s functions are continuous in C.For the region D this is equivalent to the local connectedness of ∂D inthecaseofGreen’s function, and local connectedness of the part ∂D\X,whereX is the union of the vertical rays on ∂D, if these rays are present. In terms of function h in (4.3), local connectedness is equivalent in the case of Green’s function to the condition that the set X = {x : h(x) > 0} is at most countable, and the sets X = {x : h(x) > } are finite for every >0, that is D is obtained from H by making countably many cuts, and the length of a cut tends to 0. In the case of Martin’s function, local connectedness of D means that the sets X can only accumulate to a or b. Next we discuss the condition on D which corresponds to absolute continuity of μ. We thank Misha Sodin who passed to us the contents of his conversation with Ch. Pommerenke on this subject. To state the result we first recall McMillan’s sector theorem [32], [43,Thm. 6.24]. Let f be a conformal map from H to a region G. Let sect(f)bethesetof COMB FUNCTIONS 111 points x ∈ R such that the non tangential limit f(x)existsandf(x)isthevertex of an angular sector in G. McMillan Sector Theorem [43, Theorem 6.24, p.146]. Assume that A ⊂ sect(f). Then (4.5) |A| =0 if and only if |f(A)| =0. We say that the sector condition holds in the comb region D if the function h(y) (4.6) H(x)= sup y∈(a,b) |y − x| is finite for almost all x ∈ (a, b). Geometrically it means that for almost all x in the base of the comb there exists a Stolz angle with the vertex in x. Theorem 4.1. Region D satisfies the sector condition if and only if μ is abso- lutely continuous with respect to the Lebesgue measure on R. Proof. Recall that the Lebesgue measure on the base of the comb corresponds to the harmonic measure μ on E. Assume that the sector condition holds. This means that a Borel support of the harmonic measure μ is contained in sect(φ). Let A be a Borel support of the singular component of μ. By the definition |A| = 0. Thus, by McMillan’s theorem μ(A)=0,thusμ is absolutely continuous. Conversely, assume that the harmonic measure is absolutely continuous. Recall that φ has positive imaginary part, and therefore possesses non-tangential limits for almost all x with respect to the Lebesgue measure. Therefore the limit exists for almost all x with respect to the harmonic measure as well. Example 1. There exist irregular regions with absolutely continuous measures μ. Indeed, let C be the standard Cantor set in [a, b]. Let h(x) be the characteristic function of C. Then the region generated by this comb is irregular, on other hand H(x) is finite for all x ∈ [a, b] \ C. Example 2. We give an example of a comb such that the conditions of the previous theorem do not hold, moreover H(x)=∞ for almost all x ∈ [a, b]. This comb is related to the Julia set of a polynomial T (z)=z2 − λ [46]. For λ>2there exists h0 > 0 such that the Julia set of T is the preimage of the base of the comb given in Fig. 4. Recall that almost every number x contains arbitrarily long strings of zeros in its dyadic representation, that is, for almost every x, and every non- negative integer N, there exists a string ym of 0’s and 1’s, ending with 1, such that 4 56N 7 −m −(m+N) x =(ym, 0, ..., 0, ....). Then h(ym)=2 h0,and|x − ym| =2 .Thatis N H(x) ≥ 2 h0. In fact the Lebesgue measure of the Julia set is 0, i.e. the harmonic measure is singular continuous. Note that since the boundary is locally connected the region C \ E is regular. Even stronger condition is that (4.7) h(x) < ∞, x in other words, the total length of slits is finite. This is the so-called Widom condi- tion. It appears in his studies of asymptotics for extremal polynomials associated 112 ALEXANDRE EREMENKO AND PETER YUDITSKII Figure 4. Comb related to the Julia set of T (z)=z2 − λ. with a system of curves in the complex plane. Let π1(Ω) be the fundamental group \ ∈ ∗ of the given region Ω = C E. To a fixed character α π1 (Ω) one associates the set of multi-valued (character-automorphic) analytic functions ∞ H (α)={f : f ◦ γ = α(γ)f, ∀γ ∈ π1(Ω), sup |f(z)| < ∞}. z∈Ω The region Ω is of Widom type if the space H∞(α) is non-trivial (contains a non- ∈ ∗ \ constant function) for every α π1 (Ω). A regular region Ω = C E is of Widom type if and only if (4.7) holds. For the role of this condition in the spectral theory of almost periodic Jacobi matrices see [48]. A well-known fact that the derivative of a conformal mapping on a region bounded by a rectifiable curve belongs to H1 implies that the corresponding equilibrium measure μ is absolutely continuous. 5. More general combs In this section we consider more general comb regions: those which satisfy property (ii) of the previous section. These regions D can be described as (5.1) D = {x + iy : a COMB FUNCTIONS 113 Theorem 5.1. There are the following canonical bijections between the sets Conf, R, Sub: φ → φ :Conf→ R, φ → Im φ :Conf→ Sub. Moreover, Re φ = μ +const, (2π)−1Δv = dμ. 6. Uniform approximation and extremal problems Here we consider several extremal problems whose solutions are expressed in terms of comb functions. Applications of comb functions to extremal problems begins with the work of Akhiezer and Levin [5] on extension of Bernstein’s inequality. Further applications are contained in [26, 27, 28, 29]. A survey of polynomials and entire functions of least deviation from zero on closed sets on the real line is given in [47]. Here we mention only few results. 1. Let f be an entire function of exponential type 1 satisfying |f(x)|≤1,x<0 and |f(x)|≤B, x > 0, where B ≥ 1. One looks for maximal values of |f(x)| for given x and of |f (0)|,[13]. The extremal function is expressed in terms of the −1 ≥ → MO-comb with hj =0,j<0andhj =cosh B, j 0. Let θ : H D be the conformal map onto the region (2.5), such that θ(z) ∼ z, as z →∞non-tangentially, −1 θ(0) = 0−.Setx1 = θ (ih0). Then the function ⎧ ⎨ B, x > x1, ≤ ≤ f0(x)=⎩ cos θ(x), 0 x x1, 1,x<0 gives the solution of the first extremal problem: |f(x)|≤f0(x)forf in the class | | described above, and f0(0) is the maximal value of f (0) . 2. Best uniform approximation of sgn(x) on two rays/intervals. The simplest problem of this kind is to find the best uniform approximation of sgn(x)onthe set X =(−∞, −a] ∪ [a, ∞) by entire functions of exponential type at most 1. The extremal entire function belongs to the MacLane class and its critical sequence is of the form −1+(−1)j L, j ≤ 0, (6.1) c = j 1+(−1)j L, j > 0, where L is the error of the best approximation. Unfortunately, MacLane’s functions do not have simple representations in terms of conformal mappings like (2.2) or (2.6), however in certain cases representation in terms of conformal maps of the kind described in section 5 can be obtained [15, 17, 39]. 3. Let us consider a uniform counterpart of the classical orthogonal Jacobi n polynomials. Let α, β ≥ 0 and let Jn(x; α, β)=x + ... denote the monic poly- nomial of least deviation from zero on [0, 1] with respect to the weight function xα(1 − x)β. Lemma 6.1. For non-negative α, β and an integer n α β φ x (1 − x) Jn(x)=Le , 114 ALEXANDRE EREMENKO AND PETER YUDITSKII where φ is the conformal map on the V -comb region y 3n y D = {z = x + iy : −β< <α+ n}\ {z = x + iy : = j, x ≤ 0}. π π j=0 Such polynomials turn out to be useful in the description of multidimensional polynomials of least deviation from zero [37]. As an example we formulate the following theorem. Note that in multidimensional situation an extremal polynomial is not necessarily unique. Theorem 6.2. [38] A best polynomial approximation P (z1,...,zd, z1,....zd) k1 kd l1 ≥ to the monomial z1 ...zd z1 , k1 l1, by polynomials of the total degree less than 2 2 k1 + ···+ kd + l1 in the ball |z1| + ···+ |zd| ≤ 1 canbegivenintheform zk1 ...zkd z l1 + P (z ,...,z , z ,....z ) 1 1 1 d 1 d − k − l k + ···+ k = zk1 1 zk2 ...zkd J |z |2; 1 1 , 2 d . 1 2 d l1 1 2 2 4. We finish this section with an extremal problem for entire functions of exponential type which arises in harmonic analysis [16]. For a fixed σ>0, consider the class Aσ of entire functions which can be represented in the form 1 f(z)= F (ζ)e−iζzdζ, 2π γ where F is analytic in C\[−σ, σ], F (∞)=0andγ is a closed contour surrounding once the segment [−σ, σ]. We are interested in the upper estimate of the upper density of zeros of f, n(r) d(f) = lim sup , r→∞ r where n(r) is the number of zeros, counting multiplicity in the disc {z : |z|≤r}. Theorem 6.3. For a function f ∈ Aσ, we have d(f) ≤ cσ,wherec ≈ 1.508879 is the unique solution of the equation (6.2) log( c2 +1+c)= 1+c−2 on (0, +∞). This theorem is deduced from the solution of the following extremal problem for comb regions. Among all univalent functions φ mapping H onto regions of the form D = {x + iy : y>h(|x|)} with the properties h(0) = 0,h(x) ≤ 0, φ(0) = 0,φ(iy) ∼ iy, y → +∞, find the function with the largest Re φ(1). The extremal region is described by −∞, 0 It is interesting that the same constant c as in (6.2) appears in the solution of another extremal problem [3, Appendix, 84] which has no apparent relation to Theorem 6.3. COMB FUNCTIONS 115 7. Spectral theory and harmonic analysis 1. We say that an unbounded closed set E is homogeneous if there exists η>0 such that for all x ∈ E and all δ>0, |(x − δ, x + δ) ∩ E|≥ηδ. Theorem 7.1. [50] Let θ be a conformal map from the upper half-plane H onto an MO-comb region D (Fig. 1, right). Assume that E = θ−1(R) is homogeneous. Then E is the spectrum of a periodic canonical system, i.e., there exists an integrable on [0, 1] non-negative 2 × 2 matrix function H(t) of period 1, H(t +1)= H(t), such that for an entire (transfer) matrix function T (1,z) defined by the differential system 01 (7.1) JT˙ (t, z)=zH(t)T (t, z),T(0,z)=I, J = , −10 the following relation holds (7.2) eiθ =Δ− Δ2 − 1, Δ(z):=(1/2) tr T (1,z). Moreover the parameter t in (7.1) corresponds to the “exponential type” of the matrix T (t, z) with respect to the Martin function θ, that is, log T (t, iy) (7.3) t = lim . y→+∞ Im θ(iy) The whole collection of such matrices H(t) for the given E can be parametrized by the characters of the fundamental group of the region Ω=C \ E. The condition of homogeneity of E implies that Ω = C\E is of Widom type, and thus the region DG, which corresponds to the Green function of Ω, satisfies Widom’s condition (4.7). Moreover, the so called Direct Cauchy Theorem holds in Ω. This fact plays a crucial role in the proof of Theorem 7.1. A very interesting and natural question: is it possible to characterize a Widom domain by means of geometric properties of the region D related to the Martin function of Ω? Example. A region D is defined by a system of slits forming a geometric progression k hjk = κj ,κ>0,h0 = ∞, otherwise hj = 0. The corresponding set E is homogeneous. 2. Riesz bases. A sequence of vectors (en) in a Hilbert space H is called a Riesz basis if it is complete and there exist positive constants c, C such that 8 8 2 2 8 8 2 c |an| ≤ 8 anen8 ≤ C |an| for every finite sequence (an). A long-standing problem is how to find out whether iλnx for a given sequence of real exponents (λn) the sequence e is a Riesz basis in L2(−π, π). A recent result of G¨unter Semmler gives a parametric description of such Riesz bases. We say that a sequence (dn),dn ≥ 0 satisfies the discrete Muckenhoupt condi- tion if −1 ≤ 2 dn dn C(card I) , n∈I n∈I for every interval I of integers, and some C>0. 116 ALEXANDRE EREMENKO AND PETER YUDITSKII Theorem 7.2. [44] The sequence (eiλnx) is a Riesz basis in L2(−π, π) if and only if it is the sequence of zeros of the entire function f =expφ of exponential type, where φ is a conformal map onto a V -comb with tips of the cuts hn,and exp(2hn) satisfies the discrete Muckenhoupt condition. 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MR1817660 (2002b:47097) Department of Mathematics, Purdue University West Lafayette, Indiana 47907 E-mail address: [email protected] Abteilung fur¨ Dynamische Systeme und Approximationstheorie, Johannes Kepler Universitat¨ Linz, A–4040 Linz, Austria E-mail address: [email protected] Contemporary Mathematics Volume 578, 2012 http://dx.doi.org/10.1090/conm/578/11473 Orthogonality relations for bivariate Bernstein-Szeg˝o measures Jeffrey S. Geronimo, Plamen Iliev, and Greg Knese To Francisco Marcell´an on the occasion of his 60th birthday Abstract. The orthogonality properties of certain subspaces associated with bivariate Bernstein-Szeg˝o measures are considered. It is shown that these spaces satisfy more orthogonality relations than expected from the relations that define them. The results are used to prove a Christoffel-Darboux like formula for these measures. 1. Introduction In the study of bivariate polynomials orthogonal on the bi-circle progress has recently been made in understanding these polynomials in the case when the orthog- onality measure is purely absolutely continuous with respect to Lebesgue measure of the form dσ dμ = iθ iφ 2 , |pn,m(e ,e )| where pn,m(z,w)isofdegreen in z and m in w and is stable i.e. is nonzero for |z|, |w|≤1anddσ is the normalized Lebesgue measure on the torus T2. Such measures have come to be called Bernstein-Szeg˝o measures and they played an im- portant role in the extension of the Fej´er-Riesz factorization lemma to two variables [1], [2], [4], [5]. In particular in order to determine whether a positive trigonomet- ric polynomial can be factored as a magnitude square of a stable polynomial an important role was played by a bivariate analog of the Christoffel-Darboux formula. The derivation of this formula was non trivial even if one begins with the stable polynomial pn,m,[1], [3], [4], [9]. This formula was shown to be a special case of the formula derived by Cole and Wermer [4] through operator theoretic methods. Here we give an alternative derivation of the Christoffel-Darboux formula beginning with the stable polynomial pn,m. This is accomplished by examining the orthogonality 2 properties of the polynomial pn,m in the space L (dμ). These orthogonality prop- erties imbue certain subspaces of L2(dμ) with many more orthogonality relations than would appear by just examining the defining relations for these spaces. 2010 Mathematics Subject Classification. Primary 42C05, 30E05, 47A57. Key words and phrases. Bivariate measures, Bernstein-Szeg˝o, Christoffel-Darboux, repro- ducing kernel. JSG is supported in part by Simons Foundation Grant #210169. PI is supported in part by NSF Grant #0901092. GK is supported in part by NSF Grant #1048775. c 2012 American Mathematical Society 119 120 J. GERONIMO, P. ILIEV, AND G. KNESE We proceed as follows. In section 2 we introduce the notation to be used throughout the paper and examine the orthogonality properties of the stable poly- 2 nomial pn,m in the space L (dμ). We also list the properties of a sequence of polynomials closely associated with pn,m. In section 3 we state, and in section 4, prove, one of the main results of the paper on the orthogonality of certain sub- spaces of L2(dμ). We also establish several follow-up results which are then used in section 5 to derive the Christoffel-Darboux formula. The proof is reminiscent of that given in [3]and[6]. In section 6, we study connections to the parametric moment problem. 2. Preliminaries Let pn,m ∈ C[z,w] be stable with degree n in z and m in w. We will frequently use the following partial order on pairs of integers: (k, l) ≤ (i, j)iffk ≤ i and l ≤ j. The notations , refer to the negations of the above partial order. Define ← n m pn,m(z,w)=z w pn,m(1/z,¯ 1/w¯). When we refer to “orthogonalities,” we shall always mean orthogonalities in 2 2 2 the inner product ·, · of the Hilbert space L (1/|pn,m| dσ)onT . Notice that 2 2 2 2 L (1/|pn,m| dσ) is topologically isomorphic to L (T ) but we use the different ge- ometry to study pn,m. The polynomial pn,m is orthogonal to more monomials than the one variable theory might initially suggest. More precisely, 2 2 Lemma 2.1. In L (1/|pn,m| dσ), pn,m is orthogonal to the set {ziwj :(i, j) (0, 0)} ← and pn,m is orthogonal to the set {ziwj :(i, j) (n, m)}. 2 Proof. Observe that since 1/pn,m is holomorphic in D i j i j dσ z w ,pn,m = z w pn,m(z,w) 2 2 |p (z,w)| T n,m ziwj = dσ = 0 if (i, j) (0, 0) T2 pn,m(z,w) by the mean value property (either integrating first with respect to z or w depending ← on whether i>0orj>0). The claim about pn,m follows from the observation i j ← n−i m−j z w , pn,m = pn,m,z w . m i Write pn,m(z,w)= i=0 pi(z)w . BIVARIATE BERNSTEIN-SZEGO˝ MEASURES 121 Figure 1. Orthogonalities of pn,m Since pn,m(z,w) is stable it follows from the Schur-Cohn test for stability [1] that the m × m matrix ⎡ ⎤ ⎡ ⎤ p0(z) p¯0(1/z)¯p1(1/z) ··· p¯m−1(1/z) ⎢ . ⎥ ⎢ . . ⎥ ⎢ p (z) .. ⎥ ⎢ .. . ⎥ T (z)=⎢ 1 ⎥ ⎢ ⎥ m ⎢ . . ⎥ ⎢ . ⎥ ⎣ . .. ⎦ ⎣ .. ⎦ p − (z) ··· ··· p (z) p¯ (1/z) m⎡1 0 ⎤ ⎡ ⎤ 0 p¯m(1/z) pm(z) ··· p1(z) ⎢ ⎥ ⎢ ⎥ − . . . . (2.1) ⎣ . .. ⎦ ⎣ .. . ⎦ p¯1(1/z) ··· p¯m(1/z) pm(z) is positive definite for |z| =1.Here¯pj (z)=pj(¯z). 122 J. GERONIMO, P. ILIEV, AND G. KNESE Define the following parametrized version of a one variable Christoffel-Darboux kernel p (z,w)p (1/z,η¯ ) − ←p (z,w)←p (1/z,η¯ ) (2.2) L(z,w; η)=zn n,m n,m n,m n,m 1 − wη¯ n m−1 m−1 † = z [1,...,w ]Tm(z)[1,...,η ] m −1 j = aj(z,w)¯η , j=0 where aj(z,w),j=0,...,m− 1arepolynomialsin(z,w), as the following lemma shows in addition to several other important observations. Lemma 2.2. Let pn,m(z,w) be a stable polynomial of degree (n, m).Then, (1) L is a polynomial of degree (2n, m−1) in (z,w) and a polynomial of degree m − 1 in η¯. (2) L(·, ·; η) spans a subspace of dimension m as η varies over C. (3) L is symmetric in the sense that L(z,w; η)=z2n(wη¯)m−1L(1/z,¯ 1/w¯;1/η¯), ← so ak = am−k−1. (4) L can be written as ← L(z,w; η)=pn,m(z,w)A(z,w; η)+pn,m(z,w)B(z,w; η) where A, B are polynomials of degree (n, m − 1,m− 1) in (z,w,η¯). Proof. The numerator of L vanishes when w =1/η¯,sothefactor(1− wη¯) divides the numerator. This gives (1). For (2), when |z| = 1 use equation (2.2). Since Tm(z) > 0for|z| =1,L(z,w; η) spans a set of polynomials of dimension m. For (3), this is just a computation. For (4), observe that (suppressing the dependence of p on n and m), p(z,w)p(1/z,¯ η) − ←p(z,w)←p(1/z,¯ η) zn 1 − wη¯ (2.3) m← m← m m η¯ p(z,1/η¯) − η¯ p(z,w) ← η¯ p(z,w) − η¯ p(z,1/η¯) = p(z,w) − + p(z,w) − . 6 1 74wη¯ 5 6 1 74wη¯ 5 A(z,w;η) B(z,w;η) 2 2 3. Orthogonality relations in L (1/|pn,m| dσ) Our main goal is to prove that L and a0,...,am−1 possess a great many orthog- 2 2 onality relations in L (1/|pn,m| dσ). The orthogonality relations of L are depicted in Figure 2. BIVARIATE BERNSTEIN-SZEGO˝ MEASURES 123 Figure 2. Orthogonalities of L. See Theorem 3.1 2 2 Theorem 3.1. In L (1/|pn,m| dσ),eachak is orthogonal to the set i j Ok ={z w : i>n,j<0} ∪{ziwj :0≤ j 2 2 In L (1/|pn,m| dσ), L(·, ·; η) is orthogonal to the set O ={ziwj : i>n,j<0} (3.1) ∪{ziwj : i = n, 0 ≤ j n j Ok = {z w :0≤ j 2 2 Corollary 3.2. In L (1/|pn,m| dσ), the polynomial ak is uniquely determined (up to unimodular multiples) by the conditions: i j ak ∈ span{z w :(0, 0) ≤ (i, j) ≤ (2n, m − 1)}, 124 J. GERONIMO, P. ILIEV, AND G. KNESE Figure 3. The ak are uniquely determined by the above proper- ties. See Corollary 3.2. i j ak ⊥{z w :(0, 0) ≤ (i, j) ≤ (2n, m − 1),j = k} ∪{ziwk :0≤ i ≤ 2n, i = n}, and π 2 iθ iθ dθ ||ak|| = Tk,k(e ,e ) . −π 2π (The last fact follows from Proposition 6.1, which is not currently essential.) Remark 3.3. We emphasize that (1) each ak is explicitly given from the coeffi- cients of pn,m,(2)eachak is determined by the orthogonality relations in Corollary 3.2 (depicted in Figure 3), and (3) each satisfies the additional orthogonality rela- tions from Theorem 3.1. One useful consequence of this is that the set j {z ak(z,w):j ∈ Z, 0 ≤ k 4. The proof of Theorem 3.1 We begin by writing m −1 m −1 j j A(z,w; η)= Aj (z,w)¯η B(z,w; η)= Bj (z,w)¯η . j=0 j=0 BIVARIATE BERNSTEIN-SZEGO˝ MEASURES 125 Recall equation (2.2) and Lemma 2.2 item (4). By examining coefficients ofη ¯j in L ← aj = pn,mAj + pn,mBj . Also, Aj and Bj have at most degree j in w. To see this, recall equation (2.3) and observe that m−j ← 1 − (wη¯) A(z,w; η)= p (z)¯ηj j 1 − wη¯ j which shows that Aj(z,w) has degree at most j in w (i.e. powers of w only occur next to greater powers of η). The same holds for B. Proof of Theorem 3.1. By Lemma 2.1, pn,m is orthogonal to {ziwj :(i, j) (0, 0)} and since Ak hasdegreeatmostn in z and k in w, i j pn,mAk is orthogonal to {z w :(i, j) (n, k)}. Also, ← i j pn,mBk is orthogonal to {z w :(i, j) (n, m)} ← since the orthogonality relation for pn,m (also from Lemma 2.1) is unaffected by multiplication by holomorphic monomials. ← Hence, ak = pn,mAk + pn,mBk is orthogonal to the intersection of these sets; namely, (4.1) {ziwj :(i, j) (n, k)and(i, j) (n, m)}. Since i j am−k−1 ⊥{z w :(i, j) (n, m − k − 1) and (i, j) (n, m)} ← 2n m−1 and since ak = am−k−1 = z w am−k−1(1/z,¯ 1/w¯), 2n−i m−j−1 ak ⊥{z w :(i, j) (n, m − k − 1) and (i, j) (n, m)} (4.2) = {ziwj :(n, k) (i, j)and(n, −1) (i, j)}. Hence, ak is orthogonal to the union of the sets in (4.1) and (4.2). The set in (4.2) contains {ziwj : i We now look at the space generated by shifting the ak’s by powers of z. 2 dσ Theorem 4.1. With respect to L ( 2 ), |pn,m| i span{z aj(z,w):0≤ i, 0 ≤ j 126 J. GERONIMO, P. ILIEV, AND G. KNESE Proof. Since the ak are polynomials of degree at most m − 1inw,itisclear that i i j span{z aj (z,w):0≤ i, 0 ≤ j By Theorem 3.1, the ak are orthogonal to the spaces span{ziwj : i i and since these spaces are invariant under multiplication byz ¯, the polynomials z ak are also orthogonal to these spaces for all i ≥ 0. So, i i j span{z aj(z,w):0≤ i, 0 ≤ j k span{z aj (z,w):0≤ k, 0 ≤ j Define H =span{ziwj :(0, 0) ≤ (i, j) ≤ (n, m − 1)} % span{ziwj :(0, 0) ≤ (i, j) ≤ (n − 1,m− 1)}. ← Define also the reflection H ← H =span{ziwj :(0, 0) ≤ (i, j) ≤ (n, m − 1)} % span{ziwj :(1, 0) ≤ (i, j) ≤ (n, m − 1)}. Proposition 4.2. We have the following orthogonal direct sum decompositions 2 2 in L (1/|pn,m| dσ) ?∞ k i (4.5) span{z aj (z,w):0≤ k, 0 ≤ j ∞ ? ← k j i (4.6) H1 := span{z w :0≤ k, 0 ≤ j K (z,w; z ,w ) K←(z,w; z1,w1) H 1 1 and H 1 − zz¯1 1 − zz¯1 respectively. BIVARIATE BERNSTEIN-SZEGO˝ MEASURES 127 Proof. Now H is an m dimensional space of polynomials contained in the space (4.3) of the previous theorem. In particular, (4.7) H ⊥ span{ziwj : i Since shifts of H are contained in span{ziwj :0≤ i, 0 ≤ j ← H =span{ziwj :0≤ i, 0 ≤ j ? ← j H1 = z H. j≥0 The formulas for the reproducing kernels are direct consequences of the orthog- onal decompositions (see [4] for more on this). 2 2 Lemma 4.3. In L (1/|pn,m| dσ) the reproducing kernel for H = span{ziwj :(0, 0) ≤ (i, j) (n, m)} is ←−− ←−− p (z,w)p (z ,w ) − p (z,w)p (z ,w ) n,m n,m 1 1 n,m n,m 1 1 . (1 − zz¯1)(1 − ww¯1) Proof. First, pn,m(z,w)pn,m(z1,w1) K(z,w; z1,w1)=K(z1,w1)(z,w)= (1 − zz¯1)(1 − ww¯1) 128 J. GERONIMO, P. ILIEV, AND G. KNESE is the reproducing kernel for span{ziwj :(0, 0) ≤ (i, j)} since f(z,w) dzdw f,K = p (z ,w ) (z1,w1) n,m 1 1 2 T2 pn,m(z,w) (2πi) zw(1 − zz¯ 1)(1 − ww¯ 1) f(z1,w1) = pn,m(z1,w1)=f(z1,w1) pn,m(z1,w1) by the Cauchy integral formula. On the other hand, ← ←−− p (z,w)p (z ,w ) (4.8) n,m n,m 1 1 (1 − zz¯1)(1 − ww¯1) is the reproducing kernel for (4.9) span{ziwj :(0, 0) ≤ (i, j)}%H. i j ← To see this it is enough to show that {z w pn,m :(0, 0) ≤ (i, j)} is an orthonormal i j ← basis for the space (4.9). By Lemma 2.1, z w pn,m is in the space in (4.9) for every i, j ≥ 0 and it is easy to check that these polynomials form an orthonormal set. We show that their span is dense. ← n m We may write pn,m = cz w + lower order terms with c =0,since pn,m is sta- ← n m ble. Now, let f be in the space in (4.9). If f ⊥ pn,m = cz w + lower order terms, then since f is already orthogonal to the “lower order terms” we see that f ⊥ znwm. Inductively, then, we see that assuming f ⊥ ziwj for all i ≤ N and j ≤ M but N M ← N M (i, j) =( N,M) and assuming f ⊥ z w pn,m, we automatically get f ⊥ z w N M ← since f will be orthogonal to the lower order terms in z w pn,m. Therefore, if f in i j ← (4.9) is orthogonal to {z w pn,m : i, j ≥ 0} there can be no minimal (i, j) ≥ (n, m) (in the partial order on pairs) such that f is not orthogonal to ziwj .Inparticular, f ⊥ ziwj for all i ≥ n and j ≥ m and by (4.9) f ⊥H, which forces f ≡ 0. i j ← So, {z w pn,m :(0, 0) ≤ (i, j)} is an orthonormal basis for the space in (4.9) while (4.8) is the reproducing kernel for this space. Finally, the reproducing kernel for H = span{ziwj :(0, 0) ≤ (i, j)}%(span{ziwj :(0, 0) ≤ (i, j)}%H) is the difference of the reproducing kernels we have just calculated. Namely, ←−− ←−− p (z,w)p (z ,w ) − p (z,w)p (z ,w ) n,m n,m 1 1 n,m n,m 1 1 . (1 − zz¯1)(1 − ww¯1) 5. The bivariate Christoffel-Darboux formula Set i j H1 =span{z w :0≤ i ≤ n, 0 ≤ j ≤ m − 1} % span{ziwj :0≤ i ≤ n − 1, 0 ≤ j ≤ m − 1} and ← i j H2 =span{z w :0≤ i ≤ n − 1, 0 ≤ j ≤ m} % span{ziwj , 0 ≤ i ≤ n − 1, 1 ≤ j ≤ m}. The two variable Christoffel-Darboux formula is the following. BIVARIATE BERNSTEIN-SZEGO˝ MEASURES 129 Theorem 5.1. Let pn,m be a stable polynomial. Let K1 be the reproducing ← kernel for H1 and let K2 be the reproducing kernel for H2.Then ←−− ←−− pn,m(z,w)pn,m(z1,w1) − pn,m(z,w)pn,m(z1,w1) =(1− ww¯1)K1(z,w; z1,w1)+(1− zz¯1)K2(z,w; z1,w1), Proof. Set H = span{ziwj :(0, 0) ≤ (i, j) (n, m)} i j H1 = span{z w :0≤ i, 0 ≤ j (5.2) = H1 % (H1 ∩H2) ⊂H%H2 which a fortiori implies H1 % (H1 ∩H2)=H%H2. To see this, suppose f ∈ (H%H2) % (H1 % (H1 ∩H2)). Then, f ∈H%H1.AsH1 and H2 span H,suchanf must be orthogonal to all of H and must equal 0. The reproducing kernel for the space ? j K1(z,w; z1,w1) H%H2 = z H1 is 1 − zz¯1 j≥0 from Proposition 4.2. If we interchange the roles of z and w in Proposition 4.2 we see that K2(z,w; z1,w1) 1 − ww¯1 is the reproducing kernel for H2. Finally, the reproducing kernel for H = H2 ⊕ (H%H2) can be written in two ways. On the one hand it equals ←−− ←−− p (z,w)p (z ,w ) − p (z,w)p (z ,w ) n,m n,m 1 1 n,m n,m 1 1 , (1 − zz¯1)(1 − ww¯1) but on the other it equals K (z,w; z ,w ) K (z,w; z ,w ) 2 1 1 + 1 1 1 1 − ww¯1 1 − zz¯1 by the discussion above. Equating these formulas and multiplying through by (1 − zz¯1)(1 − ww¯1), yields the desired formula. 6. Parametric orthogonal polynomials The above results also shed light on the parametric orthogonal polynomials. The following proposition shows that the inner products of a0,...,am−1 with re- spect to L2(dμθ, T) for the measures parametrized by z = eiθ ∈ T | | θ dw (6.1) dμ (w)= iθ 2 2π|pn,m(e ,w)| 130 J. GERONIMO, P. ILIEV, AND G. KNESE are trigonometric polynomials in z. Proposition 6.1. For fixed z ∈ T | | | |2 dw n (6.2) L(z,w; η) 2 =¯z L(z,η; η) T 2π|pn,m(z,w)| and as a consequence |dw| (6.3) ai(z,w)aj (z,w) 2 = Ti,j (z). T 2π|pn,m(z,w)| Proof. For z ∈ T the expression p (z,w)p (z,η) − ←p (z,w)←p (z,η) z¯nL(z,w; η)= n,m n,m n,m n,m 1 − wη¯ is the reproducing kernel/Christoffel-Darboux kernel for polynomials in w of degree 2 at most m − 1 with respect to the measure |dw|/(2π|pn,m(z,w)| ). Indeed, this is one of the main consequences of the Christoffel-Darboux formula in one variable (see [7] equation (34) or [8] Theorem 2.2.7). It is a general fact about reproducing kernels K(w, η)=Kη(w)that 2 ||K(·,η)|| = Kη,Kη = K(η, η). Using these two observations, (6.2) follows. Equation (6.3) follows from matching the coefficients of ηiη¯j in (6.2). Given Tm(z) defined in equation (2.1), set Di(θ) as the determinant of the i × i iθ submatrix of Tm(e ) obtained by keeping the first i rows and columns and set D0 = 1. We now perform the LU decomposition of Tm which because it is positive definite does not require any pivoting. Set θ θ T m−1 T (6.4) [φm−1(w),...,φ0(w)] = U(θ)[w ,...,1] , where U(θ) is the upper triangular factor obtained from the LU decomposition of Tm without pivoting. We find: Proposition . { θ }m−1 6.2 Suppose pn,m is a stable polynomial then φi (w) i=0 satisfy the relations • φθ(w) is a polynomial in w of degree i with leading coefficient, Dm−i(θ) , # i Dm−i−1(θ) θ θ θ Dm−i(θ) • φ (w)φ (w)dμ (w)=δi,j , T i j Dm−i−1(θ) which uniquely specify the polynomials. The above implies iθk θ iφ θ dθdφ e D − − (θ)φ (e )φ (eiφ) =0,k>n(m − j). m j 1 j j 2| iθ iφ |2 [−π,π]2 (2π) pn,m(e ,e ) Proof. From the definition of Tm we see that it is the inverse of the m × m moment matrix associated with dμθ(w). The first part of the result now follows from the one dimensional theory of polynomials orthogonal on the unit circle. The n(m−j) second part follows since z Dm−j (θ) is a polynomial in z. 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MR2650771 (2012c:47089) School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332– 0160 E-mail address: [email protected] School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332– 0160 E-mail address: [email protected] Department of Mathematics, University of Alabama, Box 870350, Tuscaloosa, Al- abama 35487-0350 E-mail address: [email protected] Contemporary Mathematics Volume 578, 2012 http://dx.doi.org/10.1090/conm/578/11485 Quantum walks and CMV matrices F. Alberto Gr¨unbaum Abstract. I give a very brief account of my encounter with the topics alluded to in the title. The basic idea is that by using CMV matrices one can associate a quantum walk (QW) on the non-negative integers to any probability measure on the unit circle. This development has been a joint effort with my friends M. J. Cantero, L. Velazquez and L. Moral. It is clear that this would not have happened if it were not for the enthusiasm and motivation provided by my friend Paco Marcellan. Certain aspects of this work have been described in a joint paper with Luis Velazquez, which the reader may want to consult. I include here a few points that were not discussed there. 1. Introduction and contents of the paper I heard for the first time of the CMV matrices from Paco when he paid a short visit to Berkeley a few years ago, but I did not pay as much attention as I should have. He also mentioned a new book (OPUC) by Barry Simon, and I bought a copy, but did not look at it. It was only a couple of years later, that I thought that just as Jacobi matrices play such an important role in the analysis of CLASSICAL random walks, CMV matrices should be the natural tool to study QUANTUM walks. The next step was to convince three experts on CMV matrices to join me in this adventure...... and we have had a lot of fun. The idea is that in both the classical and the quantum case we are trying to study a model of a physical process by means of SPECTRAL METHODS.This brings in (hopefully to some advantage) all the tools from real and complex analysis, functional analysis, and other parts of mathematics. I predict that this fruitful interaction will continue for many years to come. As the very recent paper [5] makes apparent, this is a two way street: QWs provide a dynamical interpretation for certain well known objects in complex analysis. After I recall the general picture given in [1, 2, 3, 4] I concentrate on the probability measure constructed by F. Riesz, [8], back in 1918. The measure on the unit circle that F. Riesz built is formally given, up to dividing by 2π,bythe 2010 Mathematics Subject Classification. Primary 81P68, 47B36, 42C05. Key words and phrases. Riesz measure, Laurent orthogonal polynomials, CMV matrices, quantum random walks. This research was supported in part by the Applied Math. Sciences subprogram of the Office of Energy Research, USDOE, under contract DE-AC03-76SF00098. c 2012 American Mathematical Society 133 134 F. ALBERTO GRUNBAUM¨ expression ∞ ∞ 1 1 k k dμ(z)= (1 + cos(4kθ)dθ = (1 + (z4 + z−4 )/2)dz/(iz) 2π 2π k=0 k=0 ∞ j =( μj z )dz/(iz) j=−∞ Here z = eiθ. If one truncates this infinite product the corresponding measure has a nice density. These approximations converge weakly to the Riesz measure which is singular continuous. The study of dynamical systems with this kind of spectrum is a rich and promising subject. The recent paper [3] gives a natural path to associate to any probability measure on the unit circle a quantum walk on the nonnegative integers Z ≥ 0andthis construction has also been pushed to quantum walks on the integers Z,see[3]. To be more precise, if one wants to fall within the traditional class of coined quantum walks considered in the literature, one needs to restrict the class of probability measures in a way that will be made clear below. We take here the attitude that for an arbitrary probability measure a slightly more general recipe for these transitions gives rise to a quantum walk. The measure considered by Riesz falls outside of the more restricted class considered so far, and is used here as an interesting example. For many remarkable properties of the Riesz measure, see [8, 6, 13]. 2. Szeg˝o polynomials and CMV matrices Let dμ(z) be a probability measure on the unit circle T = {z ∈ C : |z| =1}, 2 T and Lμ( ) the Hilbert space of μ-square-integrable functions with inner product (f,g)= f(z) g(z) dμ(z). T For simplicity we assume that the support of μ contains an infinite number of points. A very natural operator to consider in our Hilbert space is given by 2 T → 2 T (2.1) Uμ : Lμ( ) Lμ( ) f(z) −→ zf(z) 2 T Since the Laurent polynomials are dense in Lμ( ), a natural basis to obtain a ∞ matrix representation of Uμ is given by the Laurent polynomials (χj)j=0 obtained { −1 2 −2 } 2 T from the Gram–Schmidt orthonormalizalization of 1,z,z ,z ,z ,... in Lμ( ). C ∞ ∞ The matrix =(χj ,zχk)j,k=0 of Uμ with respect to (χj )j=0 has the form ⎛ ⎞ α0 ρ0α1 ρ0ρ1 0000... ⎜ ⎟ ⎜ ρ0 −α0α1 −α0ρ1 0000...⎟ ⎜ ⎟ ⎜ 0 ρ1α2 −α1α2 ρ2α3 ρ2ρ3 00...⎟ ⎜ ⎟ (2.2) C = ⎜ 0 ρ1ρ2 −α1ρ2 −α2α3 −α2ρ3 00...⎟ , ⎜ ⎟ ⎜ 00 0ρ3α4 −α3α4 ρ4α5 ρ4ρ5 ...⎟ ⎝ ⎠ 00 0ρ3ρ4 −α3ρ4 −α4α5 −α4ρ5 ...... QUANTUM WALKS AND CMV MATRICES 135 −| |2 ∞ where ρj = 1 αj and (αj)j=0 is a sequence of complex numbers such that |αj | < 1. The coefficients αj are known as the Verblunsky (or Schur, or Szeg˝o, or reflection) parameters of the measure μ, and establish a bijection between the probability measures supported on an infinite set of the unit circle and sequences of points in the open unit disk. The unitary matrices of the form above are called CMV matrices, see [10]. A very important role is played by the Carath´eodory function F of the orthog- onality measure μ, defined by t + z (2.3) F (z)= dμ(t), |z| < 1. T t − z F is analytic on the open unit disc with McLaurin series ∞ j j (2.4) F (z)=1+2 μj z ,μj = z dμ(z), T j=1 whosecoefficientsprovidethemomentsμj of the measure μ. Another useful tool in the theory of OP on the unit circle is the so called Schur function related to μ by means of F (z) through the expression f(z)=z−1(F (z) − 1)(F (z)+1)−1, |z| < 1. These functions obtained here by starting from a probability measure on T can be characterized as the analytic functions on the unit disk Δ = {z ∈C: |z| < 1} such that F (0) = 1, Re F (z) > 0and|f(z)| < 1forz ∈ Δ respectively. Starting at f0 = f, the Verblunsky coefficients αk = fk(0) can be recovered through the Schur algorithm that produces a sequence of functions fk(z)bymeans of 1 fk(z) − αk (2.5) fk+1(z)= . z 1 − αkfk(z) By using the reverse recursion 2 zfk+1(z)+αk ρk (2.6) fk(z)= = αk + 1 1+αkzfk+1(z) αk + zfk+1(z) one can obtain a continued fraction expansion for f(z). This is called a ”continued fraction-like” algorithm by Schur, [9], and made into an actual one by H. Wall in [12]. See also [10]. We will illustrate the power of this way of computing the Schur parameters by using it in our example to compute (with computer assistance in exact arithmetic) enough of them that we can formulate an ansatz as to the form of these parameters. One of the results of this paper consists of finding the Verblunsky parameters in the case of F. Riesz’s measure. In the process of finding these parameters we will need to invoke some other sequences. Some of these will be subsequences of {αj }, and some other ones will only have an auxiliary role. We will propose an ansatz for the Verblunsky parameters of the Riesz measure that have been checked so far for the first 6000 non-null Verblunsky parameters. This is enough for computational purposes concerning the related quantum walk. A proof of this ansatz deserves additional efforts. The reader may want to look at [5] for a dynamical interpretation of the Schur function introduced above. For a general introduction see [11]. 136 F. ALBERTO GRUNBAUM¨ 3. Traditional quantum walks We will consider one-dimensional quantum walks with basic states |i ⊗|↑ and |i ⊗|↓ ,wherei runs over the non-negative integers, and with a one step transition mechanism given by a unitary matrix U. This is usually done by considering a coin at each site i, as we will see below. One considers the following dynamics: a spin up can move to the right and remain up or move to the left and change orientation. A spin down can either go to the right and change orientation or go to the left and remain down. In other words, only the nearest neighbour transitions such that the final spin (up/down) agrees with the direction of motion (right/left) are allowed. This dy- namics bears a resemblance to the effect of a magnetic interaction on quantum system with spin: the spin decides the direction of motion. This rule applies to values of the site variable i ≥ 1 and needs to be properly modified at i =0toget a unitary evolution. Schematically, the allowed one step transitions are $ |i +1 ⊗|↑ with amplitude ci |i ⊗|↑ −→ 11 |i − 1 ⊗|↓ with amplitude ci $ 21 |i +1 ⊗|↑ with amplitude ci |i ⊗|↓ −→ 12 | − ⊗|↓ i i 1 with amplitude c22 where, for each i ∈ Z, i i c11 c12 (3.1) Ci = i i c21 c22 is an arbitrary unitary matrix which we will call the ith coin. If we choose to order the basic states of our system as follows (3.2) |0 ⊗|↑ , |0 ⊗|↓ , |1 ⊗|↑ , |1 ⊗|↓ ,... then the transition matrix is given below ⎛ ⎞ 0 0 c21 0 c11 ⎜ 0 0 ⎟ ⎜c22 0 c12 0 ⎟ ⎜ 1 1 ⎟ ⎜ 0 c21 00c11 ⎟ ⎜ 1 1 ⎟ U = ⎜ c22 00c12 0 ⎟ ⎜ 2 2 ⎟ ⎜ 0 c21 00c11 ⎟ ⎜ 2 2 ⎟ ⎝ c22 00c12 0 ⎠ ...... and we take this as the transition matrix for a traditional QW on the non-negative integers with arbitrary (unitary) coins Ci as in (3.1) for i =0, 1, 2,... . The reader will notice that the structure of this matrix is not too different from a CMV matrix for which the odd Verblunsky coefficients vanish. This feature will guarantee that in the CMV matrix the central 2×2 blocks would vanish identically. The CMV matrix should have real and positive entries in some of the 2×1 matrices that are adjacent to the central 2×2 blocks, and this is not true of the unitary matrix given above. In [3] one proves that this can be taken care of by an appropriate conjugation with a diagonal matrix. QUANTUM WALKS AND CMV MATRICES 137 In [3] one considers the case of a constant coin Ci for which the measure μ and the function F (z) are explicitly given. In this case, after the conjugation alluded to above, the Schur coefficients are (3.3) a, 0,a,0,a,0,a,0 ... for a value of a that depends on the coin, and the function F (z) is, up to a rotation of z, given by the function z − z−1 − 2i Im a − . (z − z−1)2 +4|a|2 − 2Rea One can see that f(z)isanevenfunctionofz, and it is easy to see that this is equivalent to requiring that the odd Verblunsky coefficients of μ should vanish. Traditional quantum walks are therefore those whose Schur function is an even function of z. It is easy to see that in terms of F (z) the restriction to a traditional quantum walk amounts to F (−z)F (z)=1. 4. Quantum walks resulting from a probability measure One of the main points of [3] was to show that the use of the measure dμ(z) allows one to associate with each state of our quantum walk a complex valued 2 T function in Lμ( ) in such a way that the transition amplitude between any two states in time n is given by an integral with respect to μ involving the corresponding functions and the quantity zn. More explicitly we have (4.1) Ψ˜ |U n|Ψ = znψ(z)ψ˜(z)dμ(z), T 2 T | | where ψ(z)istheLμ( ) function associated with the state Ψ = j ψj j .Here |j is the j-th vector of the ordered basis consisting of basic vectors as given in (15), i.e. |j>stands for a site and a spin orientation. Similarly ψ˜(z) is the function associated to the state |Ψ . This construction is now extended to the case of any transition mechanism that is cooked out of a CMV matrix as above. More explicity, we allow for the following dynamics ⎧ ⎪|i +1 ⊗|↑ with amplitude ρi+2ρi+3 ⎨⎪ |i − 1 ⊗|↓ with amplitude ρ α |i ⊗|↑ −→ i+1 i+2 ⎪|i ⊗|↑ with amplitude − α α ⎩⎪ i+1 i+2 |i ⊗|↓ with amplitude ρi+2αi+3 ⎧ ⎪|i +1 ⊗|↑ with amplitude − αi+2ρi+3 ⎨⎪ |i − 1 ⊗|↓ with amplitude ρ ρ |i ⊗|↓ −→ i+1 i+2 ⎪|i ⊗|↑ with amplitude − α ρ ⎩⎪ i+1 i+2 |i ⊗|↓ with amplitude − αi+2αi+3 The expressions for the amplitudes above are valid for i even. If i is odd then in every amplitude the index i needs to be replaced by i − 1. One can, in principle, consider even more general transitions. As long as the evolution is governed by a unitary operator with a cyclic vector there is a CMV 138 F. ALBERTO GRUNBAUM¨ matrix lurking around. In our case the basis is given directly in terms of the basic states and there is no need to look for a new basis. It is clear that all the results in [3] extend to this more general case. As long as we have a way of computing the orthogonal Laurent polynomials we get an expression for the transition amplitude, in any number of steps, between pairs of basic states. 5. The Verblunsky coefficients for Riesz’s measure It is possible in principle to compute as many Verblunsky coefficients for the Schur function f(z) as one wants. The first few ones are given below, arranged for convenience in groups of eight. We list separately the first four parameters. 1/2, −1/3, 5/8, −1/13, 1/14, −1/15, −1/4, −1/9, 1/10, −1/11, 21/32, −1/53, 1/54, −1/55, −3/52, −1/49, 1/50, −1/51, 5/56, −1/61, 1/62, −1/63, −1/20, −1/57, 1/58, −1/59, −11/48, −1/37, 1/38, −1/39, −1/12, −1/33, 1/34, −1/35, 1/8, −1/45, ... and now we come to the main point in the determination of these coefficients of the Riesz measure. We will describe a procedure that allows us to generate an infinite sequence of integers of which the first ones are given below 13, 53, 61, 37, 45, 213, 221, 197, 205, 245, 253, 229, 237, 149, 157, 133, 141,... . We denote this sequence by Ai,sothat A1 =13,A2 =53,A3 =61,... . The reader will notice that -up to a global change of sign- this is the beginning of a sequence made up of the reciprocals of the 4th, 12th, 20th,... coefficients of f(z) given above. One can see, [4], that once these coefficients are accounted for, all the remaining ones will be determined by some simple explicit formulas in terms of the Ai. For this reason we will refer to the sequence {Ai} WHICH WE ARE ABOUT TO CONSTRUCT as the backbone of the sequence {αi} we are in- terested in. 6. Building the backbone Consider sets vn defined as the ordered set of NON-NEGATIVE integers of the form −1+((−2)n−1 − 1)/3+k2n. where k runs over the integers. Here we give the first few elements of the sets v0,v1,v2,v3,...,v10. It is clear that all that we need to produce vn is its first element. v0 =1, 3, 5, 7,...; v1 =2, 6, 10, 14,...; v2 =0, 8, 16, 24,...; v3 =12, 28, 44,...; v4 =4, 36, 68,...; v5 =52, 116, 180,...; v6 =20, 148, 276,...; v7 = 212, 468, 724,...; v8 =84, 596, 1108,...; v9 = 852, 1876, 2900,...; v10 = 340, 2388, 4436,...; QUANTUM WALKS AND CMV MATRICES 139 These sets vj , j ≥ 0, are disjoint and their union gives all integers. I give below a simple proof of this observation of mine, supplied by my friend B. Poonen. Take any polynomial ax + b where a and b are nonzero integers such that a does not divide b. This insures that the solution to ax + b = 0 is not an integer. Then one can define the set vj to be the set of integers x such that the exponent of 2 in the factorization of ax + b is j. These sets obviously form a partition of the set of integers. In our case we are dealing with the special case 3x +1. We now return to the construction started above. Define now, for n ≥ 4, n−4 5 cn =8+((−2) − 1)2 /3 so that the values of c4,c5,c6,c7,... are given by (6.1) 8, −24, 40, −88, 168, −344, 680, −1368, 2728,... a sequence whose first differences are of the form −((−2)n+1),n=4, 5, 6,... . For each pair j, n define wj,n as the number of elements in the sequence vj that are not larger than n. Finally define, for n ≥ 0, ∞ un = c4+kwk,n. k=0 Notice that by definition this is a finite sum since, for a given n the expression wk,n ∞ vanishes if k is large enough. Notice that j=0 wj,n = n and that wj,0 is zero for all j, making u0 =0. Consider now the sequence obtained adding the value 13 to each un, i.e., (6.2) 13 + u0, 13 + u1, 13 + u2, 13 + u3,... . We now take this explicitly defined sequence as the basic ingredient to construct the Verblunsky coefficients of the Riesz measure. The reader will have no difficulty verifying that this sequence starts as the one we are calling the backbone of {αi}, namely Ai,i≥ 1. At the time when this paper is being written this is a conjecture that has been verified for the first six thousand coefficients, but a general proof is still missing. In [4] there is an explict formula for wj,n. There one also finds a different way to express the Ai giving a nice description of the limit set of the Verblunsky coefficients. 7. Building up the chain In this section we describe a different way to build our sequence. This may have independent interest if one is trying to connect this work to other constructions. The construction below is not included in [4]. Consider the set of symbols A,B,C,D,E,F,G,H,I,... that will be used to obtain the integers that make up the backbone of our chain. The construction goes as follows: At location 0 place the symbol A and copy this symbol over all the integers at every location that differs from 0 by a multiple of 2, i.e., ...,−6, −4, −2, 0, 2, 4, 6,... . 140 F. ALBERTO GRUNBAUM¨ This is exactly the set v0 defined earlier. So far we have ...,,A,,A,,A,,A,,A,,A,,A,,A,,A,,A,,A,,A,,A,,A,,A¯ ,,A,,A,,A,,... . We have used the symbol A¯ to denote the location 0. Take the first unoccupied site to the LEFT of 0, namely −1=0− 1 and place there the symbol B. Copy this symbol over all the integers that differ from −1by a multiple of 4, i.e., at ...,−13, −9, −5, −1, 3, 7, 11, 15,... . Thissetisexactlythesetv1. Having inserted the new symbol we have ...,,B,A,,A,B,A,,A,B,A,,A,B,A,,A,B,A,,A,B,A,,A,B,A,,A,B,A,,A¯ ,B,A,,A,B,A,,... . Take the first unoccupied site to the RIGHT of 0, namely 1 = −1+2 and place there the symbol C.Copythissymboloveralltheintegersthatdifferfrom1bya multiple of 8, i.e., at ...,−23, −15, −7, 1, 9, 17, 25,... . This set is, once again, the set v2 introduced above and by now we get ...,,A,B,A,C,A,B,A,,A,B,A,C,A,B,A,,A,B,A,C,A,B,A,,A,B,A,C,A,B,A,,...¯ . It should be clear how this construction can be continued: by going alternatively to the left and the right we keep running into the sets vj defined above. At each location belonging to the new set vj we copy a new symbol from the ordered set of symbols A,B,C,D,E,F,G,H,I,.... We now associate with the symbol A the value c4 = 8 given earlier, with the symbol B the value c5 = −24 given earlier, etc. For any value of n starting at n = 1 we look at the positive part of our chain, starting at location 1 for which the symbol is C and going all the way to the symbol Sn at location n, i.e. we will have the string of symbols C,A,B,A,E,A,B,A,C,A,B,A,D,A,B,A,C,A,B,A,...,Sn Finally, we simply add up all the values assigned to these symbols and get an integer. We claim that if we add 13 to this integer we get the corresponding value of our sequence. This is exactly the meaning of the formula ∞ un = c4+kwk,n k=0 given in the previous section. References [1]M.J.Cantero,F.A.Gr¨unbaum, L. Moral and L. Vel´azquez, One-dimensional quantum walks with one defect, Reviews in Mathematical Physics, 24, 125002 (2012). [2]M.J.Cantero,F.A.Gr¨unbaum, L. Moral and L. Vel´azquez, The CGMV method for quantum walks, to appear in Quantum Information Processing, Y. Shikano (editor). [3]M.J.Cantero,F.A.Gr¨unbaum, L. Moral and L. Vel´azquez, Matrix valued Szeg˝o polynomials and quantum random walks, Commun. Pure Applied Math. 58 (2010) 464–507. MR2604869 (2011b:81137) QUANTUM WALKS AND CMV MATRICES 141 [4] F. A. Gr¨unbaum and L. Velazquez, The Quantum Walk of F. Riesz FoCAM 2011, Budapest, Hungary, to be published by the London Mathematical Society Lecture Notes Series. See also arXiv:1111.6630v1 (math-ph). [5]F.A.Gr¨unbaum, L. Velazquez, R. Werner and A. Werner, Recurrence for discrete time unitary evolutions, to appear in Commun. Math. Phys. [6] Y. Katznelson, An introduction to harmonic analysis, John Wiley & Sons, 1968. MR0248482 (40:1734) [7] N. Konno, Quantum walks, in Quantum Potential Theory, U. Franz, M. Sch¨urmann, editors, Lecture notes in Mathematics 1954, Springer Verlag, Berlin Heidelberg, 2008. MR2463710 (2010i:81174) [8] F. Riesz, Uber¨ die Fourierkoeffizienten einer stetigen Funktion von beschr¨ankter Schwankung, Math. Z. 18 (1918) 312–315. MR1544321 [9] I. Schur, Uber¨ Potenzreihen die im Innern des Einheitskreises beschr¨ankt sind,J.Reine Angew. Math., 147 (1916) 205–232 and 148 (1917) 122–145. [10] B. Simon, Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory, AMS Colloq. Publ., vol. 54.1, AMS, Providence, RI, 2005. MR2105088 (2006a:42002a) [11] G. Szeg˝o, Orthogonal Polynomials, 4th ed., AMS Colloq. Publ., vol. 23, AMS, Providence, RI, 1975. MR0372517 (51:8724) [12] H. Wall, Continued fractions and bounded analytic functions,BAMS,50 (1944) 110-119. MR0010211 (5:262e) [13] A. Zygmund, Trigonometric series, 2nd ed. , Cambridge University Press, 1959. MR0107776 (21:6498) Department of Mathematics, University of California, Berkeley, California 94720 Contemporary Mathematics Volume 578, 2012 http://dx.doi.org/10.1090/conm/578/11470 Discrete beta ensembles based on Gauss type quadratures D. S. Lubinsky Abstract. Let μ be a measure with support on the real line and n ≥ 1, β>0. In the theory of random matrices, one considers a probability distribution on the eigenvalues t1,t2,...,tn of random matrices, of the form P(n) | |β β (μ; t1,t2,...,tn)=C V (t1,t2,...,tn) dμ (t1) ...dμ(tn) , where C is a normalization constant, and V (t1,t2,...,tn)= (tj − ti) . 1≤i 1. Introduction Let μ be a finite positive Borel measure on the real line with infinitely many points in the support, and all finite moments. Let β>0 and n ≥ 2. The β-ensemble, with temperature 1/β, associated with the measure μ places a probability distribution on the eigenvalues t1,t2,...,tn of an n by n Hermitian matrix, of the form P(n) β (μ; t1,t2,...,tn) 1 β (1.1) = |V (t1,t2,...,tn)| dμ (t1) ···dμ (tn) , Zn where − j−1 (1.2) V (t1,t2,...,tn)= (tj ti)=det ti 1≤i,j≤n 1≤i 1991 Mathematics Subject Classification. Primary 41A10, 41A17, 42C99; Secondary 33C45. Key words and phrases. Random matrices. Research supported by NSF grant DMS1001182 and US-Israel BSF grant 2008399. c 2012 American Mathematical Society 143 144 D. S. LUBINSKY and β (1.3) Zn = ··· |V (t1,t2,...,tn)| dμ (t1) ···dμ (tn) . These ensembles arise in scattering theory in mathematical physics. Their analysis has generated interest amongst mathematicians and physicists for decades [2], [3], [4]. One of the important statistics is the m-point correlation function Rm,β (μ; y ,y ,...,y ) n 1 2 m n! = ··· P(n) (μ; y ,y ,...,y ,t ,...,t ) dμ (t ) ···dμ (t ) (n − m)! β 1 2 m m+1 n m+1 n # # n! ··· |V (y ,y ,...,y ,t ,...,t )|β dμ (t ) ···dμ (t ) = # 1 # 2 m m+1 n m+1 n . − β (n m)! ··· |V (t1,t2,...,tn)| dμ (t1) ···dμ (tn) (1.4) It can be used to study local spacing properties of eigenvalues, and local density of eigenvalues. For example, if m = 2, and B ⊂ R is measurable, then 2,β Rn (μ; t1,t2) dμ (t1) dμ (t2) B B is the expected number of pairs (t1,t2) of eigenvalues, with both t1,t2 ∈ B. The best understood case is β =2[2], where there are close connections to the the theory of orthogonal polynomials associated with the measure μ.Thecasesβ = 1 and β = 4 are also well understood [3], [4], although the analysis is far more complicated. For Jacobi weights, one can use the Selberg integral to partly analyze general β. For the case where β is the square of an integer, some analysis has been undertaken by Chris Sinclair [17]. A recent breakthrough by Borgade, Erd˝os, and Yau [1] gives a new approach to handling β-ensembles for varying weights of the form e−nV with V convex and real analytic. In this paper, we show that when we take μ to be a Gauss type quadra- ture measure, then we can explicitly evaluate the correlation function, and hence analyze universality limits for sequences of such measures, at least for the case β>1. Define orthonormal polynomials n pn (x)=γnx + ··· ,γn > 0, n =0, 1, 2, ···, satisfying the orthonormality conditions pjpkdμ = δjk. DISCRETE BETA ENSEMBLES BASED ON GAUSS TYPE QUADRATURES 145 Throughout we use μ to denote the Radon-Nikodym derivative of μ.The nth reproducing kernel for μ is n −1 Kn (μ, x, y)= pk (x) pk (y) . k=0 Its normalized cousin is 1/2 1/2 K˜n (μ, x, y)=μ (x) μ (y) Kn (μ, x, y) . The nth Christoffel function is n −1 2 λn (μ, x)=1/Kn (μ, x, x)=1/ pj (x) . j=0 When it is clear that the measure is μ, we’ll omit the μ, just writing λn (x) and Kn (x, y). Recall that given any real ξ with (1.5) pn−1 (ξ) =0 , there is a Gauss quadrature including ξ as one of the nodes: n (1.6) Pdμ= λn (μ, xjn) P (xjn) j=1 ≤ − { }n { }n for P of degree 2n 2. We shall usually order xjn j=1 = xjn (ξ) j=1 in increasing order; in Section 3, we shall adopt a different notation, setting x0n = ξ.The{xjn} are zeros of ψn (t, ξ)=pn (ξ) pn−1 (t) − pn−1 (ξ) pn (t) . In the special case that pn (ξ) = 0, these are the zeros of pn, and the precision of the quadrature is actually 2n − 1. Note that when pn−1 (ξ)=0,thereis still a quadrature like (1.6), but involving n − 1 points, namely the zeros of pn−1, and exact for polynomials of degree ≤ 2n − 3. We define the discrete measure μn by n (1.7) fdμn = λn (μ, xjn) f (xjn) . j=1 Equivalently, n (1.8) μn = λn (μ, xjn) δxjn, j=1 where δxjn denotes a Dirac delta at xjn. Note that μn depends on ξ, but we shall not explicitly display this dependence. Our basic identity is: 146 D. S. LUBINSKY Theorem 1.1. Let μ be a measure on the real line with infinitely many points in its support, and all finite power moments. Let β>0, n>m≥ 1; let ξ ∈ R satisfy (1.5),andμn be the discrete measure defined by (1.8).For any real y1,y2,...,ym, m,β Rn (μn; y1,y2,...,ym) m β−1 1 = λ (μ, x ) m! n jkn 1≤j ,j ,...,j ≤n k=1 % ⎡1 2 m ⎤% % %β % Kn (μ, xj n,y1) ... Kn (μ, xj n,ym) % % ⎢ 1 1 ⎥% × % ⎣ . .. . ⎦% (1.9) %det . . . % . % % Kn (μ, xjmn,y1) ... Kn (μ, xjmn,ym) Remark. ≤ ≤ (a) Suppose that yk = xjkn,1 k m, for some distinct 1 ≤ j1,j2,...,jm ≤ n. Then the above reduces to m m,β −1 Rn (μn; xj1n,xj2n,...,xjmn)= λn (μ, xjkn) . k=1 (b) If m = 1, we see that n 1,β β−1 | |β Rn (μn; y)= λn (μ, xjn) Kn (μ, y, xjn) . j=1 n −1 β = λn (μ, xjn) |jn (y)| , j=1 where {jn} are the fundamental polynomials of Lagrange interpolation for {xjn}. (c) When β = 2, this reduces to a familiar identity in random matrix theory: Corollary 1.2. m,2 m,2 Rn (μn; y1,y2,...,ym)=Rn (μ; y1,y2,...,ym) (1.10) =det[Kn (μ, yi,yj)]1≤i,j≤m . The representation in Theorem 1.1 lends itself to asymptotics: let sin πt (1.11) S (t)= πt denote the sinc kernel. Recall that a compactly supported measure μ is said to be regular in the sense of Stahl, Totik, and Ullman,orjustregular,ifthe leading coefficients {γn} of its orthonormal polynomials satisfy 1/n 1 lim γn = . n→∞ cap (supp [μ]) DISCRETE BETA ENSEMBLES BASED ON GAUSS TYPE QUADRATURES 147 Here cap(supp [μ]) is the logarithmic capacity of the support of μ. We recall only a very simple criterion for regularity, namely a version of the Erd˝os- Tur´an criterion: if the support of μ consists of finitely many intervals, and μ > 0 a.e. with respect to Lebesgue measure in that support, then μ is regular [18, p. 102]. There are many deeper criteria in [18]. We also need the density ωJ of the equilibrium measure for a compact set J.ThusωJ (x) dx is the unique probability measure that minimizes the energy integral 1 log dν (s) dν (t) |s − t| amongst all probability measures ν with support in J [13], [14]. In the special case J =[−1, 1], ω (x)= √ 1 . J π 1−x2 Theorem 1.3. Let μ be a regular measure with compact support J.Let I be a compact subinterval of J such that μ is absolutely continuous in an open interval I1 containing I. Assume that μ is positive and continuous in I1, and moreover, that either (1.12) sup p < ∞, n L∞(I1) n≥1 or ∞ (1.13) sup n λn L∞(J) < . n≥1 Fix ξ ∈ I,andforn ≥ 1, assume (1.5) holds. Let μn include the point ξ as one of the quadrature points. Then for β ≥ 2 and real a ,a ,...,a , 1 2 m, μ (ξ) m a a lim Rm,β μ ; ξ + 1 ,...,ξ+ m →∞ n n n nωJ (ξ) nωJ (ξ) nωJ (ξ) ∞ % % 1 % %β (1.14) = %det [S (a − j )] % . m! i k 1≤i,k≤m j1,j2···jm=−∞ For 1 <β<2, the same result holds if we assume (1.12) and the additional restriction n 1 −1 − (1.15) λn (μ, xkn) = o n 1 β/2 . k=1 Remarks. (a) We can also write the limit as 1 a a lim Rm,β μ ; ξ + 1 ,...,ξ+ m →∞ m n n n Kn (μ, ξ, ξ) K˜n (μ, ξ, ξ) K˜n (μ, ξ, ξ) ∞ % % 1 % %β (1.16) = %det [S (a − j )] % , m! i k 1≤i,k≤m j1,j2···jm=−∞ because, uniformly in compact subsets of I1, 1 lim K˜n (x, x)=ωJ (x) . n→∞ n 148 D. S. LUBINSKY (b) If the support of μ is the interval [−1, 1] and μ satisfies the Szego con- dition 1 log μ (x) √ dx > −∞, 2 −1 1 − x while in some open subinterval I2 of (−1, 1), μ is absolutely continuous, μ is bounded above and below by positive constants, and μ satisfies the condition % % % %2 %μ (t) − μ (θ)% % % dt < ∞ t − θ uniformly in I1, then (1.12) holds (cf. [5, p. 246, Thm. V.4.4]). In particular, this holds for Jacobi and generalized Jacobi weights. The bound (1.12) is also known for exponential weights that violate Szeg˝o’s condition [7]. (c) The global condition√ (1.13) is satisfied if, for example, the support is [−1, 1] and μ (x) ≤ C/ 1 − x2 for a.e. x ∈ (−1, 1). In fact, as we show in Section 3, one can replace (1.12) and (1.13) by the more implicit condition (which they both imply) (1.17) sup λn (t) |Kn (x, t)|≤C, n ≥ 1. t∈J,x∈I2 Here I2 is a compact subinterval of I1 that contains I in its interior. (d) (1.15) places severe restrictions on the measure μ, especially near the endpoints of the support. But some such restriction may well be necessary. It seems that universality is most universal for the “natural” case β =2. (e) When β = 2, the last right-hand side reduces to a familiar universality limit: Corollary . 1.4 μ (ξ) m a a lim Rm,2 μ ; ξ + 1 ,...,ξ+ m →∞ n n n nωJ (ξ) nωJ (ξ) nωJ (ξ) − =det[S (ai aj)]1≤1,j≤m . Of course, this last limit has been established under much more general conditions elsewhere, using special techniques available for β =2[9], [10], [16], [21]. For β = 4, the form of the universality limit differs from the standard one for β = 4 as the determinant of a 2 by 2 matrix involving S and its derivatives and integrals [3, p. 142]. It remains to be seen if (1.14) coincides with that form. We prove Theorem 1.1 and Corollary 1.2 in Section 2, and Theorem 1.3 and Corollary 1.4 in Section 3. Throughout C, C1,C2,... denote positive constants independent of n, x, t, that are different in different occurrences. 2. Proof of Theorem 1.1 and Corollary 1.2 We shall often use (2.1) Kn (μ, xjn,xkn)=0,j= k. DISCRETE BETA ENSEMBLES BASED ON GAUSS TYPE QUADRATURES 149 We also use the notation r =(r ,r ,...,r ) and s =(s ,s ,...,s ) ¯n 1 2 n ¯n 1 2 n and D ((r1,r2,...,rn) , (s1,s2,...,sn)) = D (r , s )=det[K (r ,s )] ¯⎡n ¯n n i j 1≤i,j≤n ⎤ K (r ,s ) K (r ,s ) ... K (r ,s ) ⎢ n 1 1 n 1 2 n 1 n ⎥ ⎢ Kn (r2,s1) Kn (r2,s2) ... Kn (r2,sn) ⎥ (2.2) =det⎢ . . . . ⎥ . ⎣ . . .. . ⎦ Kn (rn,s1) Kn (rn,s2) ... Kn (rn,sn) Lemma 2.1. β ··· |V (t1,t2,...,tn)| dμn (t1) ···dμn (tn) n 1−β/2 −β (2.3) =(γ0 ···γn−1) n! λn (μ, xkn) . k=1 Proof. We see by taking linear combinations of columns that ··· γ0γ1 γn−1V (t1,...,tn)=det[pk−1 (tj)]1≤j,k≤n . Then as the determinant of a matrix equals that of its transpose, 2 2 (γ γ ···γ − ) V (t ,...,t ) =det[p − (t )] det [p − (t )] 0 1 n 1 1 n k 1 j 1≤j,k≤n k 1 1≤k,≤n n =det pk−1 (tj) pk−1 (t) k=1 1≤j,≤n (2.4) =det[Kn (tj,t)]1≤j,≤n . Let (j1,...,jn) be a permutation of (1, 2,...,n). Then 2 ··· − [γ0γ1 γn 1V (xj1n,...,xjnn)] =det[Kn (xjin,xjn)]1≤i,≤n n = Kn (xjn,xjn) , j=1 by (2.1). Note that this is independent of the permutation (j1,...,jn). Then by definition of μn, and as V (t1,...,tn) vanishes unless all its entries are distinct, β β [γ0γ1 ···γn−1] ··· |V (t1,t2,...,tn)| dμn (t1) ···dμn (tn) n n n n β/2 ··· ··· 2 2 = λn (xjkn) (γ0γ1 γn−1) (V (xj1n,...,xjnn)) j1=1 j2=1 jn=1 k=1 j1,j2,...,jn distinct 150 D. S. LUBINSKY n n n n n β/2 = ··· λn (xkn) Kn (xkn,xkn) k=1 j1=1 j2=1 jn=1 k=1 j ,j ,...,j distinct 1 2 n n 1−β/2 = n! λn (xkn) . k=1 Recall that we use the abbreviations λn (x)forλn (μ, x), and Kn (x, y) for Kn (μ, x, y). We shall do this fairly consistently in the proof of Lemma 2.2 and Theorem 1.1. Lemma 2.2. Let m ≥ 2 and y1,y2,...,ym ∈ R.Letjm+1,jm+2,...,jn be distinct indices in {1, 2,...,n}.Let{j1,j2,...,jm} = {1, 2,...,n}\ {jm+1,...,jn}.Then D y ···y ,x ,x ,...,x , y ···y ,x ,x ,...,x 1 m jm+1 n jm+2n jnn 1 m jm+1n jm+2n jnn m n = λn (xjkn) Kn (xjkn,xjkn) k=1 k=m+1 ⎛ ⎡ ⎤⎞ 2 Kn (xj1n,y1) ... Kn (xj1n,ym) ⎜ ⎢ . . . ⎥⎟ × ⎝det ⎣ . .. . ⎦⎠ . Kn (xjmn,y1) ... Kn (xjmn,ym) (2.5) Proof. We use the reproducing kernel and Gauss quadrature in the form n (2.6) Kn (yk,u)= λn (xjin) Kn (yk,xjin) Kn (xjin,u) . i=1 ! " ∈ Substituting (2.6) with u y1,y2,...,ym,xjm+1n,...,xjnn in the first m rows of D = ··· ··· D y1 ym,xjm+1n,xjm+2n,...,xjnn , y1 ym,xjm+1n,xjm+2n,...,xjnn and then extracting each of the m sums, gives n n n m D = ··· λ x K y ,x n jik n n k jik n i1=1 i2=1 im=1 k=1 DISCRETE BETA ENSEMBLES BASED ON GAUSS TYPE QUADRATURES 151 ⎡ ⎤ K x ,y ... K x ,y K x ,x ... K x ,x ⎢ n ji n 1 n ji n m n ji n jm+1n n ji n jnn ⎥ ⎢ 1 1 1 1 ⎥ ⎢ ...... ⎥ ⎢ ...... ⎥ ⎢ ...... ⎥ ⎢ ⎥ ⎢ Kn xj n,y1 ... Kn xj n,ym Kn xj n,xj n ... Kn xj n,xj n ⎥ ⎢ im im im m+1 im n ⎥ × det ⎢ ⎥ . ⎢ K x ,y ... K x ,y K x ,x ... K x ,x ⎥ ⎢ n jm+1n 1 n jm+1n m n jm+1n jm+1n n jm+1n jnn ⎥ ⎢ ⎥ ⎢ . . . . ⎥ ⎢ ...... ⎥ ⎣ ...... ⎦ Kn (xjnn,y1) ... Kn (xjnn,ym) Kn xjnn,xjm+1n ... Kn (xjnn,xjnn) We see that this determinant vanishes unless {i1,i2,...,im} = {1, 2,...,m} (for if not, two rows of the determinant are identical). When {i1,i2,...,im} = {1, 2,...,m}, the determinant in the last equation becomes ⎡ ⎤ Kn xj n,y1 ... Kn xj n,ym 0 ... 0 ⎢ i1 i1 ⎥ ⎢ ⎥ ⎢ ...... ⎥ ⎢ ...... ⎥ ⎢ . . . . ⎥ ⎢ ⎥ ⎢ Kn xj n,y1 ... Kn xj n,ym 0 ... 0 ⎥ det ⎢ im im ⎥ ⎢ ⎥ ⎢ Kn xj n,y1 ... Kn xj n,ym Kn xj n,xj n ... 0 ⎥ ⎢ m+1 m+1 m+1 m+1 ⎥ ⎢ . . . . ⎥ ⎣ ...... ⎦ ...... Kn (xj n,y1) ... Kn (xj n,ym)0... Kn (xj n,xj n) ⎡ n n ⎤ n n K x ,y ... K x ,y ⎢ n ji n 1 n ji n m ⎥ ⎢ 1 1 ⎥ n ⎢ . . . ⎥ =det⎢ . . . ⎥ Kn xj n,xj n ⎣ . . . ⎦ k k k=m+1 Kn xj n,y1 ... Kn xj n,ym im im ⎡ ⎤ Kn xj n,y1 ... Kn xj n,ym ⎢ 1 1 ⎥ n ⎢ . . . ⎥ = εσ det . . . Kn xj n,xj n , ⎣ . . . ⎦ k k k=m+1 Kn (xjmn,y1) ... Kn (xjmn,ym) where εσ denotes the sign of the permutation σ = {i1,i2,...,im} of {1, 2,...,m}, that is ij = σ (j)foreachj,1≤ j ≤ m.Then m n D = λn (xjkn) Kn (xjkn,xjkn) k=1 ⎡ k=m+1 ⎤ Kn (xj1n,y1) ... Kn (xj1n,ym) ⎢ . . . ⎥ × det ⎣ . .. . ⎦ Kn (xjmn,y1) ... Kn (xjmn,ym) m × εσ Kn yk,xjσ(k)n σ k=1 m n = λn (xjkn) Kn (xjkn,xjkn) k=1 k=m+1 ⎛ ⎡ ⎤⎞ 2 Kn (xj1n,y1) ... Kn (xj1n,ym) ⎜ ⎢ . . . ⎥⎟ × ⎝det ⎣ . .. . ⎦⎠ . Kn (xjmn,y1) ... Kn (xjmn,ym) 152 D. S. LUBINSKY m,β Proof of Theorem 1.1. We first deal with the numerator in Rn defined by (1.4). Using the definition (1.8) of μn, the identity (2.4), and then Lemma 2.2, β β (2.7) I =(γ0γ1 ···γn−1) ··· |V (y1,y2,...,ym,tm+1,...,tn)| dμn × (t ) ···dμ (t ) m+1 n n n n n ··· = λn (xjkn) jm%+1=1 jn=1 k=m+1 % × % D y1,...,ym,xjm+1n,xjm+2n,...,xjnn , % %β/2 y1,...,ym,xj n,xj n,...,xj n % m+1 m+2 n n n n ··· = λn (xjkn) jm+1=1 jn=1 k=m+1 ··· jm$ +1 jn distinct m n × λn (xjkn) Kn (xjkn,xjkn) k=1 k=m+1 ⎛ ⎡ ⎤⎞ 2) Kn (xj1n,y1) ... Kn (xj1n,ym) β/2 ⎜ ⎢ . . . ⎥⎟ × ⎝det ⎣ . .. . ⎦⎠ Kn (xjmn,y1) ... Kn (xjmn,ym) Here {j1,j2,...,jm} = {1, 2,...,n}\{jm+1,...,jn}. Because of the sym- metry in this last expression, it is the same as it would be if j1 DISCRETE BETA ENSEMBLES BASED ON GAUSS TYPE QUADRATURES 153 $ ) 1−β/2 β−1 (n − m)! n m = λ (x ) λ (x ) m! n kn n jkn k=1 1≤j ,j ···j ≤n k=1 % ⎡ 1 2 m ⎤% % %β % Kn (xj n,y1) ... Kn (xj n,ym) % % ⎢ 1 1 ⎥% × % ⎣ . .. . ⎦% %det . . . % . % % Kn (xjmn,y1) ... Kn (xjmn,ym) Then (1.4), Lemma 2.1, and our definition (2.7) of I give m,β R (μn; y1,y2,...,ym) n # # n! ··· |V (y ,y ,...,y ,t ,...,t )|β dμ (t ) ···dμ (t ) = # 1 # 2 m m+1 n n m+1 n n − β (n m)! ··· |V (t1,t2,...,tn)| dμn (t1) ···dμn (tn) n! I = # # − β β (n m)! (γ0 ···γn−1) ··· |V (t1,t2,...,tn)| dμn (t1) ···dμn (tn) β−1 1 m = λ (x ) m! n jkn 1≤j ,j ···j ≤n k=1 % ⎡1 2 m ⎤% % %β % Kn (xj n,y1) ... Kn (xj n,ym) % % ⎢ 1 1 ⎥% × % ⎣ . .. . ⎦% %det . . . % . % % Kn (xjmn,y1) ... Kn (xjmn,ym) 2 Proof of Corollary 1.2. For β =2,|V (y1,y2,...,ym,tm+1,...,tn)| is a polynomial of degree ≤ 2n − 2intm+1,tm+2,...,tn. Similarly for 2 |V (t1,...,tn)| . Then the Gauss quadrature formula gives the first equality in (1.10). Next for β = 2, the right-hand side of (1.9) becomes 1 m λ (μ, x ) m! n jkn 1≤j ,j ···j ≤n k=1 % 1 2⎡ m ⎤% % %2 % Kn (μ, xj n,y1) ... Kn (μ, xj n,ym) % % ⎢ 1 1 ⎥% × % ⎣ . .. . ⎦% %det . . . % % K (μ, x ,y ) ... K (μ, x ,y ) % n jmn 1 n jmn m 1 = ··· det [K (μ, t y )]2 dμ (t ) dμ (t ) ···dμ (t ) . m! n i, j 1 2 m By the equality part of Theorem 1.1 in [11], this last expression equals det [Kn (yi,yj)]1≤i,j≤m. 3. Proof of Theorem 1.3 and Corollary 1.4 We begin with 154 D. S. LUBINSKY Lemma 3.1. Assume that μ satisfies the hypotheses of Theorem 1.3.Let I2 be a compact subinterval of I1.Then (a)Uniformlyforξ ∈ I2, and uniformly for a, b in compact subsets of the real line, a b Kn μ, ξ + ˜ ,ξ+ ˜ (3.1) lim Kn(ξ,ξ) Kn(ξ,ξ) = S (a − b) , →∞ n Kn (μ, ξ, ξ) (b)Uniformlyforx ∈ I2, (3.2) lim nλn (μ, x)=πμ (x) /ωJ (x) . n→∞ Moreover, there exist C1,C2 > 0 such that for n ≥ 1 and all x ∈ I2, (3.3) C1 ≤ nλn (μ, x) ≤ C2. (c) There exists C3,C4 > 0 such that for all n, j with xjn,xj−1,n ∈ I2, (3.4) C4/n ≥ xjn − xj−1,n ≥ C3/n. (d) Fix ξ ∈ I1 and {xjn} = {xjn (ξ)}. Order them in the following way: (3.5) ··· (3.6) lim (xjn − ξ) K˜n (ξ,ξ)=j. n→∞ Proof. (a) This follows from results of Totik [21, Theorem 2.2]. (b) The first part (3.2) also follows from the result of Totik [21, Theorem 2.2]. The second part follows from the extremal property of Christoffel functions, and comparison with, e.g. the Christoffel function for the Legendre weight - see [12, p. 116]. (c) We need the fundamental polynomial kn of Lagrange interpolation that satisfies kn (xjn)=δjk. One well known representation of kn, which follows from the Christoffel- Darboux formula, is (3.7) kn (x)=Kn (xkn,x) /Kn (xkn,xkn) . Let I3 be a compact subinterval of I1 that contains I2 in its interior. Then 1=jn (xjn) − jn (xj−1,n) − = jn (ξ)(xjn xj−1,n) (3.8) ≤ Cnsup |jn (t)| (xjn − xj−1,n) , t∈I3 by Bernstein’s inequality. Here for t ∈ I3, our bounds on the Christoffel function, and Cauchy-Schwarz give |jn (t)| = λn (μ, xkn) |Kn (t, xjn)| C ≤ λ (μ, x )(K (t, t))1/2 (K (x ,x ))1/2 ≤ n = C, n kn n n jn jn n DISCRETE BETA ENSEMBLES BASED ON GAUSS TYPE QUADRATURES 155 by (3.3). Then the right-hand inequality in (3.4) follows from (3.8). The left-hand inequality follows easily from the Markov-Stieltjes inequalities [5, p. 33] xjn − xj−1,n ≤ λn (xj−1,n)+λn (xjn) . (d) The method is due to Eli Levin [8], in a far more general situation than that considered here. We do this first for j = 1. By (c), and (3.3), an x1n = ξ + , K˜n (ξ,ξ) where an ≥ 0 and an = O (1). We shall show that (3.9) lim an =1. n→∞ { } Let us choose a subsequence an n∈S with lim an = a. n→∞,n∈S Because of the uniform convergence in (a), K (x ,ξ) 0 = lim n 1n →∞ ∈S n ,n Kn (ξ,ξ) an Kn ξ + ˜ ,ξ sin πa = lim Kn(ξ,ξ) = S (a)= . n→∞,n∈S Kn (ξ,ξ) πa It follows that a is a positive integer. If a ≥ 2, then as S (t) changes sign at 1, the intermediate value theorem shows that there will be a point bn yn = ξ + , K˜n (ξ,ξ) with yn ∈ (ξ,x1n), with bn → 1, and Kn (yn,ξ) = 0. This contradicts that x1n is the first zero to the right of ξ. Thus necessarily a =1.Asthis is independent of the subsequence, we have (3.9), and hence the result for j = 1. The general case of positive can be completed by induction on j. Negative j is similar. We now analyze the main part of the sum in (1.9): in the sequel, the sets I1,I2,I3 are as above. Lemma 3.2. Assume that for 1 ≤ k ≤ m, an,k (3.10) yk = yk (n)=ξ + , K˜n (ξ,ξ) where for 1 ≤ k ≤ m, lim an,k = ak, n→∞ and a1,a2,...,am are fixed. Then for each fixed positive integer L, ,m β−1 λn (xjkn) lim k=1 →∞ m n Kn (ξ,ξ) |j1|,|j2|,...,|jm|≤L 156 D. S. LUBINSKY % ⎡ ⎤% % %β % Kn (xj n,y1) ... Kn (xj n,ym) % % ⎢ 1 1 ⎥% × % ⎣ . .. . ⎦% %det . . . % % K (x ,y ) ... K (x ,y ) % n jmn 1 n jmn m β (3.11) = |det (S (ji − ak))| . |j1|,|j2|,...,|jm|≤L Proof. Note that for each fixed j, Lemma 3.1(b), (d), and the conti- nuity of μ give K (x ,x ) (3.12) n jn jn =1+o (1) . Kn (ξ,ξ) Moreover, j+o(1) an,k Kn ξ + ,ξ+ Kn (xjn,yk) K˜n(ξ,ξ) K˜n(ξ,ξ) (3.13) = = S (j − ak)+o (1) , Kn (ξ,ξ) Kn (ξ,ξ) because of the uniform convergence in Lemma 3.1(a). Hence, for each m−tuple of integers j1,j2,...,jm, ⎡ ⎤ Kn (xj1n,y1) ... Kn (xj1n,ym) 1 ⎢ . . . ⎥ m det ⎣ . .. . ⎦ Kn (ξ,ξ) Kn (xjmn,y1) ... Kn (xjmn,ym) − (3.14) =det[S (ji ak)]1≤i,k≤m + o (1) . Then using (3.12), − ⎡ ⎤ m β 1 β λ x Kn (xj1n,y1) ... Kn (xj1n,ym) n jkn ⎢ ⎥ k=1 ⎢ . . . ⎥ m det ⎣ . .. . ⎦ Kn (ξ, ξ) . . |j1|,|j2|,...,|jm|≤L Kn (xjmn,y1) ... Kn (xjmn,ym) ⎡ ⎤ β Kn (x ,y1) ... Kn (x ,ym) ⎢ j1n j1n ⎥ −mβ ⎢ . . . ⎥ =(1+o (1)) Kn (ξ, ξ) det ⎣ . . . ⎦ , . . . |j1|,|j2|,...,|jm|≤L Kn (xjmn,y1) ... Kn (xjmn,ym) and the lemma follows from (3.14). Now we estimate the tail. We assume (3.10) throughout. First we deal with the (known) case β =2: Lemma 3.3. As L →∞, ,m λn (xj n) % % k % %2 k=1 % % → (3.15) TL,2 = m det [Kn (xjin,yk)]1≤i,k≤m 0. Kn (ξ,ξ) (j1,j2,...,jm): maxi|ji|>L DISCRETE BETA ENSEMBLES BASED ON GAUSS TYPE QUADRATURES 157 Proof. Recall that from Theorem 1.1 and Corollary 1.2, ,m λn (xj n) % % 1 k % %2 k=1 % % m det [Kn (xjin,yk)]1≤i,k≤m m! ≤ ≤ Kn (ξ,ξ) 1 j1 ,j2,...,jm n K (y ,y ) =det n i j , Kn (ξ,ξ) 1≤i,j≤m and that from Corollary 1.4 below, ∞ % % 1 % %2 %det [S (a − j )] % m! i k 1≤i,k≤m j1···jm=−∞ − =det[S (ai aj)]1≤i,j≤m . (Formally, we have not yet proven this, but of course it is independent of the hypotheses here.) Now we split up the sum in the first of these identities, take limits as n →∞, and use Lemma 3.2 for β = 2, as well as the limit (3.1), which ensures that Kn (yi,yj) lim det =det[S (ai − aj)] ≤ ≤ . n→∞ 1 i,j m Kn (ξ,ξ) 1≤i,j≤m Lemma 3.4. Assume the hypotheses of Theorem 1.3,exceptfor(1.12) and (1.13). Then for n ≥ 1,andt ∈ J, 2 ≤ 2 2 (3.16) pn (t) C pn−2 (t)+pn−1 (t) . Proof. We shall show below that γ − (3.17) inf n 1 ≥ C. n γn Once we have this, we can apply the three term recurrence relation in the form γn−1 γn−2 pn (x)=(x − bn) pn−1 (x) − pn−2 (x) , γn γn−1 ' ( and the fact that {|b |} and γn−1 are bounded above, (for J = supp [μ] n γn is compact) to deduce (3.16). We turn to the proof of (3.17). From the confluent form of the Christoffel-Darboux formula, we have γn−1 Kn (xjn ,xjn )= pn−1 (xjn) pn (xjn) . γn Let I4 be a non-empty compact subinterval of I3. By the spacing estimate (3.4), there are at least C4n zeros xjn ∈ I4,so 158 D. S. LUBINSKY % % ≤ γn−1 % % C4n λn (xjn) Kn (xjn ,xjn )= λn (xjn) pn−1 (xjn) pn (xjn) γn xjn∈I4 xjn∈I4 ⎛ ⎞ ⎛ ⎞ 1/2 1/2 ≤ γn−1 ⎝ 2 ⎠ ⎝ 2⎠ λn (xjn) pn−1 (xjn) λn (xjn) pn (xjn) . γn j xjn∈I4 (3.18) The first quadrature sum is 1. By a theorem of P. Nevai [12, p. 167, Thm. 23], followed by Bernstein’s inequality, the second sum may be es- timated as ⎛ ⎞ 1/2 1/2 ⎝ 2⎠ ≤ 2 λn (xjn) pn (xjn) C pn (t) dt I xjn∈I4 4 1/2 ≤ Cn p2 (t) dt ≤ Cn, n I4 recall that μ is bounded above and below in I3.WealsouseI4 and I4 to denote nested intervals containing I4 but inside I3. Substituting in (3.18) gives (3.17). Next we handle the case β>2: Lemma 3.5. Assume all the hypotheses of Theorem 1.3,except(1.12) and (1.13). Instead of those, assume (3.19) sup λn (t) |Kn (x, t)|≤C, n ≥ 1, t∈J,x∈I2 where I2 is a compact subinterval of I1 containing I in its interior. Let β>2.ThenasL →∞, (3.20) ,m β−1 λn (xj n) % % k % %β k=1 % % → TL,β = m det [Kn (xjin,yk)]1≤i,k≤m 0. Kn (ξ,ξ) (j1,j2,...,jm): maxi|ji|>L In particular, (3.19) holds when (1.12) or (1.13) holds. Proof. We see that (3.21) ⎧ ⎫ β−2 ⎨⎪ % %⎬⎪ m % % TL,β ≤ TL,2 max λn (xj n) %det [Kn (xj n,yk)] % , ⎩⎪ k i 1≤i,k≤m ⎭⎪ (j1,j2,...,jm): k=1 maxi|ji|>L DISCRETE BETA ENSEMBLES BASED ON GAUSS TYPE QUADRATURES 159 where by Lemma 3.3, TL,2 → 0asL →∞.Next,ifσ denotes a permutation of {1, 2,...,m}, we see that % % m % % % % λn (xjkn) det [Kn (xjin,yk)]1≤i,k≤m k=1 m % % ≤ % % λn (xjkn) Kn xjkn,yσ(k) σ k=1 m ≤ m! sup λn (t) |Kn (t, y)| ≤ C, t∈J,y∈I2 by our hypothesis (3.19). Combined with (3.21), this gives the result. We turn to proving (3.19) under (1.12) or (1.13). Recall that I ⊂ I2 ⊂ I3 ⊂ I1. If firstly t ∈ I3 and x ∈ I2, 1/2 1/2 λn (t) |Kn (x, t)|≤λn (t) Kn (x, x) Kn (t, t) ≤ C, by (3.3). In the sequel, we let 2 2 An (t)=pn (t)+pn−1 (t) . From the Christoffel-Darboux formula, 1/2 1/2 γ − A (t) A (x) (3.22) |K (x, t)|≤ n 1 n n . n γ |x − t| ' ( n Here γn−1 is bounded as μ has compact support. If next, t ∈ I and γn 3 x ∈ I2,wehave|x − t|≥C,so | |≤ 1/2 1/2 λn (t) Kn (x, t) Cλn (t) An (t) An (x) . Here by Lemma 3.4, λn (t) An (t) ≤ Cλn (t) An−1 (t) ≤ C,so λ (t) |K (x, t)|≤C (λ (t) A (x))1/2 . n n n n # If (1.12) holds, then An (x) ≤ C, while λn (t) ≤ dμ, so (3.19) follows. If instead (1.13) holds, then −1 1/2 λn (t) |Kn (x, t)|≤C n An (x) −1 1/2 ≤ C n Kn+1 (x, x) ≤ C, by (3.3). Thus in all cases, we have (3.19). The case β<2 is more difficult: Lemma 3.6. Assume all the hypotheses of Theorem 1.3, including (1.12) and (1.15).Letβ<2.ThenasL →∞, (3.20) holds. Proof. Each term in TL,β has the form ,m β−1 λn (xj n) % % k % %β k=1 % % m det [Kn (xjin,yk)]1≤i,k≤m Kn (ξ,ξ) 160 D. S. LUBINSKY m % % C − (3.23) ≤ λ (x )β 1 %K x ,y %β , nm n jkn n jkn σ(k) σ k=1 ∈ Here the sum is over all permutations σ.Iffirstxjkn I3, then by the estimate (3.3) for λn, and by (3.22), % % 1 β−1 % %β λn (xjkn) Kn xjkn,yσ(k) n β/2 β/2 C An (xj n) An y ≤ % k % σ(k) nβ % − %β xjkn yσ(k) C ≤ % % , % − % β n xjkn yσ(k) by our bound (1.12) on pn. Here, recalling (3.10), % % % % % a % % − % % − − n,σ(k) % xjkn yσ(k) = %xjkn ξ % K˜n (ξ,ξ) |j | max |a | ≥ C k − C i i , 1 n 2 n by (3.4) and (3.3). It follows that there exists B>0 depending only on maxi |ai| such that for |jk|≥B, % % |j | %x − y % ≥ C k . jkn σ(k) 3 n | |≥ ∈ In particular, B is independent of L. Then for jk B, and xjkn I3, % % 1 β−1 % %β ≤ C (3.24) λn (xjkn) Kn xjkn,yσ(k) β . n (1 + |jk|) Now if |jk|≤B, we can just use our bounds (3.3) on λn and Cauchy-Schwarz to deduce that % % 1 β−1 % %β ≤ 1 β ≤ C λn (xjkn) Kn xjkn,yσ(k) C β n β . n n (1 + |jk|) ∈ Thus again (3.24)% holds, so we% have (3.24) for all jk with xjkn I3.Nextif ∈ % − % ≥ xjkn / I3,then xjkn yσ(k) C,so % % 1 − λ (x )β 1 %K x ,y %β n n jkn n jkn σ(k) C − ≤ λ (x )β 1 Aβ/2 (x ) Aβ/2 y n n jkn n jkn n σ(k) C − ≤ λ (x )β 1 Aβ/2 (x ) , n n jkn n jkn DISCRETE BETA ENSEMBLES BASED ON GAUSS TYPE QUADRATURES 161 by (1.12). Note that there is no dependence on σ in the bound in this last inequality nor in (3.24). Then ⎛ ⎞ − 1 − T ≤ C ⎝ (1+|j |) β⎠ λ (x )β 1Aβ/2 (x ) . L,β k n n jkn n jkn x ∈I ∈ (j1,j2,...,jm): jkn 3 xjkn /I3 maxi|ji|>L We can bound this above by a sum of m terms, such that in the kth term, the index jk exceeds L in absolute value, while all remaining indices may assume any integer value. As each such term is identical, we may assume that j1 is the index with |j1|≥L, and deduce that ⎛ ⎞ − 1 − T ≤ C ⎝ (1 + |j |) β + λ (x )β 1 Aβ/2 (x )⎠ L,β 1 n n j1n n j1n |j1|≥L xj n∈/I3 ⎛ 1 ⎞ m−1 ∞ × ⎝ | | −β 1 β−1 β/2 ⎠ (1 + j ) + λn (xjn) An (xjn) . −∞ n j= xjn∈/I3 −1 2 − β Here by H¨older’s inequality with parameters p = β and q = 1 2 , 1 − λ (x )β 1 Aβ/2 (x ) n n j1n n j1n ∈ xj1n /I3 1 − ≤ (λ (x ) A (x ))β/2 λ (x )β/2 1 n n j1n n j1n n j1n j ⎛1 ⎞ ⎛ ⎞ β/2 1−β/2 C − ≤ ⎝ λ (x ) A (x )⎠ ⎝ λ (x ) 1⎠ . n n j1n n j1n n j1n j1 j1 Here by Lemma 3.4, ≤ ≤ λn (xj1n) An (xj1n) C λn (xj1n) An−1 (xj1n) 2C, j1 j1 while ⎛ ⎞ 1−β/2 ⎝ −1⎠ λn (xj1n) = o (n) j1 by our hypothesis (1.15). Thus 1−β TL,β ≤ C L + o (1) , and the lemma follows. 162 D. S. LUBINSKY Proof of Theorem 1.3. This follows directly from Lemmas 3.2, 3.5 and 3.6: we can choose L so large that the tail in Lemma 3.5 or 3.6 is as small as we please. Note that in (3.10), an,k a˜n,k yk = ξ + = ξ + , K˜n (ξ,ξ) nωJ (ξ) wherea ˜n,k → ak as n →∞, in view of (3.2). This allows us to prove the universality limit in both the forms (1.14) and (1.16). Proof of Corollary 1.4. We have to prove that ∞ − 2 − det [S (ai jk)]1≤i,k≤m = m!det[S (ai ak)]1≤i,k≤m . j1,j2···jm=−∞ We use the identity [19, p. 91] ∞ S (a − k) S (b − k)=S (a − b) . k=−∞ The left-hand side is ∞ − 2 det [S (ai jk)]1≤i,k≤m j1,j2···jm=−∞ ∞ m = εσεη S aσ(k) − jk S aη(k) − jk σ,η j1,j2···jm=−∞ k=1 m ∞ = εσεη S aσ(k) − jk S aη(k) − jk σ,η k=1 jk=−∞ m = εσεη S aσ(k) − aη(k) σ,η k=1 m = εσεη S aj − aη◦σ−1(j) , σ,η j=1 where σ−1 denotes the inverse permutation of σ.Now[6, p. 189, p. 190] εσεη = εη◦σ−1 , and we may replace the sum over all permutations ω = η ◦ σ−1 by a sum over all permutations ω, so we continue this as m = εω S aj − aω(j) σ ω j=1 − = m!det[S (ai aj)]1≤i,j≤m . DISCRETE BETA ENSEMBLES BASED ON GAUSS TYPE QUADRATURES 163 References [1] P. Bourgade, L. Erdos,˝ H-T Yau, Universality of General β−Ensembles,manuscript. [2] P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Ap- proach, Courant Institute Lecture Notes, Vol. 3, New York University Pres, New York, 1999. 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Sinclair, Ensemble Averages when β is a Square Integer, Monatshefte f¨ur Math- ematik, 166 (2012), 121–144. [18] H. Stahl and V. Totik, General Orthogonal Polynomials, Cambridge University Press, Cambridge, 1992. MR1163828 (93d:42029) [19] F. Stenger, Numerical Methods Based on Sinc and Analytic Functions, Springer, New York, 1993. MR1226236 (94k:65003) [20] V. Totik, Asymptotics for Christoffel Functions for General Measures on the Real Line, J. d’ Analyse de Mathematique, 81(2000), 283–303. MR1785285 (2001j:42021) [21] V. Totik, Universality and fine zero spacing on general sets,Arkivf¨or Matematik, 47(2009), 361–391. MR2529707 (2010f:42055) School of Mathematics, Georgia Institute of Technology, Atlanta, Geor- gia 30332-0160 E-mail address: [email protected] Contemporary Mathematics Volume 578, 2012 http://dx.doi.org/10.1090/conm/578/11474 Heine, Hilbert, Pad´e, Riemann, and Stieltjes: John Nuttall’s work 25 years later Andrei Mart´ınez-Finkelshtein, Evgenii A. Rakhmanov, and Sergey P. Suetin This paper is dedicated to the 60th Birthday of Francisco (Paco) Marcell´an Abstract. In 1986 J. Nuttall published a paper in Constructive Approxima- tion, where with his usual insight he studied the behavior of the denominators (“generalized Jacobi polynomials”) and the remainders of the Pad´e approxi- mants to a special class of algebraic functions with 3 branch points. 25 years later we try to look at this problem from a modern perspective. On one hand, the generalized Jacobi polynomials constitute an instance of the so- called Heine-Stieltjes polynomials, i.e. they are solutions of linear ODE with polynomial coefficients. On the other, they satisfy complex orthogonality re- lations, and thus are suitable for the Riemann-Hilbert asymptotic analysis. Along with the names mentioned in the title, this paper features also a spe- cial appearance by Riemann surfaces, quadratic differentials, compact sets of minimal capacity, special functions and other characters. 1. Pad´e approximants to algebraic functions John Nuttall, whose name appears in the title along with such distinguished, actually illustrious, colleagues, initiated the study of convergence of Pad´e approx- imants for multivalued analytic functions on the plane. Obviously, he was not the first to consider this problem; the best known result in this sense is a theorem of Markov (or Markoff) [22], see also [27], which assures the locally uniform conver- gence of diagonal Pad´e approximants to Markov functions: if dσ(t) σ (z):= , z − t 1991 Mathematics Subject Classification. Primary 42C05; Secondary 41A20, 41A21, 41A25. Key words and phrases. Pad´e approximation, algebraic functions, Heine-Stieltjes polyno- mials, Van Vleck polynomials, WKB analysis, asymptotics, zero distribution, Riemann-Hilbert method. The first author was partially supported by Junta de Andaluc´ıa, grant FQM-229 and the Excellence Research Grant P09-FQM-4643, as well as by the research projects MTM2008-06689- C02-01 and MTM2011-28952-C02-01 from the Ministry of Science and Innovation of Spain and the European Regional Development Fund (ERDF). The third author was partially supported by the Russian Fund for Fundamental Research grant 11-01-00330, and the program “Leading Scientific Schools of the Russian Federation”, grant NSh-8033.2010.1. c 2012 American Mathematical Society 165 166 A. MART´INEZ-FINKELSHTEIN, E. A. RAKHMANOV, AND S. P. SUETIN where σ is a positive measure compactly supported on R with an infinite number of points of increase, then the diagonal Pad´e approximants [n/n]σ (see the definition in Section 2) to σ , which coincide with the approximants of the Chebyshev or J- continued fraction for this function, converge to σ uniformly on compact subsets of the complement to the convex hull of the support of σ, and the convergence holds with a geometric rate. This theorem applies in particular to functions as 1 or (z2 − 1)1/2 − z. (z2 − 1)1/2 In the same vein, Dumas [13] studied the case of the function of the form a + a + a + a f(z)=((z − a )(z − a )(z − a )(z − a ))1/2 − z2 + 1 2 3 4 z, 1 2 3 4 2 with points aj ∈ C in general position and the branch of the square root selected in such a way that f is bounded at infinity. Dumas observed that the poles of the Pad´e approximants to f can be dense in C. However, Nuttall was the first to abandon the real line completely and start a convergence theory in a truly complex situation. From the Dumas’ work it was clear that in a general situation we can no longer expect uniform convergence1. The appropriate notion is the convergence in capacity [33, 34, 45], introduced first by Pommerenke [32] and developed further independently by Gonchar and Nuttall2. This is an analogue of convergence in measure, where the Lebesgue or plane measure is replaced by the logarithmic capacity of the set. Still, the question about the domain of convergence (even in capacity) remained: if the approximated function f has a multi-valued analytic continuation to C ex- cept for a finite number of branch points, then the single-valued Pad´e approximants [n/n]f cannot converge to f in this whole domain. They must “choose” the appro- priate region of convergence where f is single-valued too, and the boundary of this region should attract a sufficient number of poles of [n/n]f . In [28, 29, 31] Nuttall generalized Markov’s theorem by considering a class −1/2 of hyperelliptic functions of the form r1 + r2h ,whereh is a polynomial of even degree and simple poles, and rj are holomorphic functions (it was extended later to meromorphic functions in the work of Stahl [42] and Suetin [47]). For these functions he found the domain where the convergence takes place: it is a complement to a system of arcs determined by the location of the branch points. Nuttall characterized this set as having a minimal logarithmic capacity among all other systems of cuts making the approximated function single-valued in their complement (see e.g. Figure 1). In [28] Nuttall conjectured also that this result is valid for any analytic function on C with a finite number of branch points. The complete proof of this conjecture was given, even in a greater generality, by H. Stahl in a series of papers [35, 36, 37, 40, 43], under the only assumption that 1We cannot expect uniform convergence even along subsequences, as it was shown in [4, 5, 21]. 2Pommerenke knew about Nuttall’s work, inspired by some lemma of Hadamard about cover- ing sets of small values of polynomials, where convergence in measure was used. Pommerenke sub- stituted convergence in measure by that in capacity, which was slightly stronger, but more impor- tantly, it simplified the proof considerably. The same year Gonchar, who deepened Pommerenke’s result by bringing in the connection between convergence speed and the single–valuedness of the approximated function, cited Pommerenke’s paper in [15]. We are very grateful to Herbert Stahl for this historical clarification. JOHN NUTTALL’S WORK 25 YEARS LATER 167 2.0 2.0 1.5 1.5 1.0 1.0 0.5 0.5 0.0 0.0 0.5 0.5 1.0 1.0 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 −3/7 Figure 1. Poles of π71 and π72 for f(z)=(z +1.2) (z − 0.7 − 1.75i)1/7(z − 1 − 0.8i)2/7. Clearly visible is a “spurious” pole (left) or a distortion of the location of the poles (right). the singularities of the function f form a polar set, i.e. a set of logarithmic capacity zero, see Theorem 2.1 below. Stahl also characterized the analytic arcs forming the boundary of this domain as trajectories of a rational quadratic differential with poles at the singularities of f. They are also a case of the so-called Boutroux curves, see e.g. [3]. The general results of Nuttall and Stahl (and also of Gonchar and Rakhmanov [16, 17]) deal essentially with convergence in capacity and weak (equivalently, n- th root) asymptotics of the denominators and residues of the Pad´e approximants. However, strong or Szeg˝o-type asymptotics is extremely interesting, at least in order to clarify the behavior of the spurious poles. As it was mentioned, poles that appear within the domain of convergence in capacity and that receive the name (coined by G. Baker in the 1960s) of spurious, floating or wandering poles [44, 47], can become the main obstacle for the uniform convergence. For some classes of elliptic and hyperelliptic functions [18, 47, 48], the dynamics of the spurious poles is completely determined by the properties of the Riemann surface underlying the approximated function. Moreover, in the elliptic case [48]thereis only one wandering pole, which greatly simplifies the description of the asymptotic behavior of the Pad´e approximants. The results in [48] were obtained when the approximated functions could be represented as Cauchy integrals supported on the critical trajectory of certain qua- dratic differential with four poles when this trajectory consists of two disconnected components; the analogue for the Chebotarev set was treated in [2]. In this paper we analyze the strong asymptotics of the Pad´e denominators and the residues of the Pad´e approximants for a canonical class of algebraic functions – a generalization of the Jacobi weight on [−1, 1], revisiting and extending the results of J. Nuttall in [30]. We look at this problems from two different perspectives, which give us two formally distinct answers. One of the main goals is to understand the relation between these two asymptotic expressions, in order to get new insight into the nature of this problem and the methods we have used. 168 A. MART´INEZ-FINKELSHTEIN, E. A. RAKHMANOV, AND S. P. SUETIN At the final stage of the preparation of this manuscript we learned about the closely related work [1] where the authors also apply one of the techniques used by us to the asymptotic analysis of the Pad´e denominators in a slightly more gen- eral situation. Our works, although close in spirit, have a number of substantial differences, and thus are rather complementary than overlapping. 2. Statement of the problem Let aj , j =1,...,p, be distinct and in general, non-collinear points on the complex plane C, A := {a1,...ap}.LetA(C \A) denote the set of functions f holomorphic at infinity and such that f can be continued analytically (as a multivalued function) to the whole C \A.Adiagonal Pad´e approximant to f ∈ A(C \A) is a rational function πn =[n/n]f = Pn/Qn of type (n, n), that is, such that both Pn,Qn ∈ Pn (where Pn denotes the class of algebraic polynomials of degree ≤ n), which has a maximal order of contact with f at infinity: 2n+1 (2.1) f(z) − πn(z)=O 1/z as z →∞. This condition may be impossible to satisfy, but following Frobenius, we can obtain the coefficients of P and Q as a solution to the linear system n n n+1 (2.2) Rn(z):=Qn(z)f(z) − Pn(z)=O 1/z as z →∞,Qn ≡ 0. Equations (2.2) form an undetermined homogeneous linear system. Although the solution of (2.2) is not unique, the Pad´e approximant (rational function) πn = Pn/Qn is. Hereafter, (Pn,Qn) will always stand for the unique pair of relatively prime polynomials determining πn,andQn is taken monic. The degree of Qn could be strictly JOHN NUTTALL’S WORK 25 YEARS LATER 169 In other words, Γ is given by the level curves of Re G(·, ∞)=0that join zeros of A or V . Alternatively, Γ is made of the closure of critical trajectories of the quadratic differential −(V/A)(z)dz2. (ii) The normalized zero counting measure for Qn converges weakly to the equi- librium measure λΓ of Γ, given by 1 V (z) (2.5) dλ (z)= dz, Γ πi A(z) with an appropriate choice of the branch of the square root. (ii) With an appropriate normalization of Rn, 1 cap log |R (z)| −→ − Re G(z,∞),n→∞, n n cap where −→ denotes convergence in capacity in C \ Γ. Remark 2.2. Depending on the function f (and the corresponding class Kf ), Stahl’s compact is not necessarily connected. For instance, already for p =4it can be a tree (for f(z)=A1/4(z)) or a union of two analytic arcs (when f(z)= 1/2 A (z)). In the class Kf Stahl’s compact however is completely characterized by its S-property,namely ∂ ∂ (Re G(z,∞)) = (Re G(z,∞)) ,z∈ Γ◦, ∂n− ∂n+ ◦ where n± are the normal vectors to Γ .TheS-property and its generalizations play a crucial role in many branches of analysis and mathematical physics, see e.g. [1, 3, 17, 23, 25, 26]. In this paper we concentrate on a canonical example of a function from A(C \ A). Namely, let αj ∈ R \ Z, j =1,...,p, be such that α1 + ···+ αp =0.We additionally assume that no proper subset of αj ’s adds up to an integer, so that the corresponding Stahl compact Γ ∈ Kf is a continuum. This is a sufficient condition for an underlying Riemann surface having the maximal genus (see below). However, for more precise hypotheses, see Assumption 3.3. The simplest non-trivial example of this situation is when p = 3, when Γ is star-shaped (Chebotarev continuum). As it follows from the works of Stahl, such “stars” along with analytic curves are the main building blocks for a generic Γ. Let p ∞ fk (2.6) f(z)= (z − a )αj =1+ , j zk j=1 k=1 where the expansion is convergent in the neighborhood O := {z ∈ C : |z| > 1/2 maxj |aj |} of infinity. We will agree in denoting by f the branch of the square root in O such that f 1/2(∞)=1. As it was mentioned above, our main goal is to find the strong asymptotics of the Pad´e denominators Qn,asn →∞. Wederivethisasymptoticsusingtwo complementary methods. The first one, developed in Section 3, is based on the differential equation satisfied by f and is a combination of the original ideas of Nuttall from [30] with some new developments in the asymptotic theory of gener- alized Heun differential equations. The second method is the non-linear steepest 170 A. MART´INEZ-FINKELSHTEIN, E. A. RAKHMANOV, AND S. P. SUETIN descent analysis of Deift and Zhou (see e.g. [11]) based on the matrix Riemann- Hilbert problem [14] solvable in terms of Qn and Rn. In Section 5 we apply it exclusively to the case of p =3. These two methods provide formally different expressions for the leading term of asymptotics of the Pad´e denominators and Pad´e residues. We find their comparison in Section 6 very illuminating. The form of the asymptotics for Qn and Rn was actually conjectured by Nuttall in [29] in terms of a function solving certain scalar boundary value problem. We show also that our results match Nuttall’s conjecture, see Section 6. 3. Heine and Stieltjes, or asymptotic analysis based on the Liuoville-Green approximation The key observation is that function f in (2.6) is semiclassical: it satisfies the ODE p p f (z) αj B = = (z),A(z):= (z − a ),B∈ P − . f(z) z − a A j n 2 j=1 j j=1 It can be proved by standard methods (see e.g. [30]) that as a consequence, the Pad´e denominators Qn,thePad´e numerators Pn, and the remainders Rn satisfy the Laguerre equations: p−2 Theorem 3.1. For each normal index n there exist polynomials hn(x)=x + 2p−4 ···∈Pp−2 and Dn(x)=x + ···∈P2p−4, such that − − − (3.1) Ahny +(A hn Ahn Bhn) y n(n +1)Dny =0 is solved by Rn, Qnf and Pn. Remark 3.2. InthecaseofQn, the ODE is of the form − − Ahnyn +(A hn Ahn + Bhn) yn n(n +1)Dnyn =0. Let us use the notation zk,n for the zeros of the polynomials hn from Theorem 3.1: p −2 (3.2) hn(z)= (z − zk,n). k=1 In order to simplify the situation and concentrate on the main ideas we impose the following assumptions on the zeros of hn: Assumption 3.3. There exists a constant M>0 such that for all sufficiently large n, | |≤ | |≥ − (3.3) zk,n M and AVnhn(zk,n) C, k =1,...,p 2. In other words, all zeros zk,n of hn belong to the disk |z|≤M, and they stay away 3 from the zeros of AVn and from each other. 3 The general case requires the spherical normalization for hn and has to be treated separately. We avoid further discussion of this situation for the sake of simplicity. JOHN NUTTALL’S WORK 25 YEARS LATER 171 Observe that the second part of this assumption is completely innocent: in a general case, we will have a reduction in genus and in a number of cycles in the basis of the underlying Riemann surface, see below. From Assumption 3.3 it follows that for all sufficiently large n the zeros of Dn lie in the disk |z|≤2M, see e.g. [23], so that that the set en := {z ∈ C : AhnDn =0} is uniformly bounded. Define in O, z Dn Hn(z):= (t) dt, a1 Ahn where the branch is chosen such that Hn(z) = log z + O(1) as z →∞.Itcanbe extended as an analytic and multivalued function to the whole C. The polynomial solution Pn of(3.1)isknownasaHeine-Stieltjes polynomial, while the corresponding coefficient Dn in (3.1) is called a Van Vleck polynomial,see e.g. [23, 24]. Theorem 2.1 from [24] gives a global description of the trajectories 2 2 of the quadratic differential (Hn) (z)dz . In particular, it is a quasi-closed differ- ential with one trajectory emanating from each zero of A and ending at infinity. Combining techniques from [24]and[30]weget Theorem 3.4. For any a ∈Athere exists a progressive path4 γ = γ(a),starting at a point z0 ∈ O and returning back to z0, which is homotopic in C\en to a contour γ+ with a ∈ Int(γ+) and en \{a}⊂Ext(γ+). For any such a progressive path γ we have for z ∈ γ ∪ O, 3/4 1/2 hn f − (3.4) R (z)=C (z) e (n+1/2)Hn(z) (1 + δ (z)) . n n,1 1/4 1 (ADn) If for ρ>0, dist(z,en) ≥ ρ,thenn|δ1(z)| is uniformly bounded by a constant depending on ρ. Remark 3.5. This formula should be understood in the following way: the right hand side is chosen for z ∈ O according to the branch of Hn described above, and then both the left and the right hand sides are continued analytically along γ. In this way this formula may be extended from progressive paths to rectangles in − the ζ = e Hn(z) plane. Constant C = Cn,1 in (3.4) depends on the normalization of Rn. Formula (3.4) is not totally satisfactory, since it has a number of undetermined parameters. Our next task is to clarify their behavior. Using Assumption 3.3 and compactness argument we can choose a subsequence Λ={nk}⊂N such that hn → h, Dn → D, as n ∈ Λ, so that by (3.4), z 1 cap D(t) log |Rn(z)| −→ − Re dt, n ∈ Λ, n a1 Ah(t) for z ∈ O. Let Γ be the Stahl’s compact associated with f, i.e. f is holomorphic in C \ Γ, and Γ has the minimal capacity in the class A(C \A). Then, it follows from 4 Progressive path γ means that Re Hn(z) is non increasing along γ. 172 A. MART´INEZ-FINKELSHTEIN, E. A. RAKHMANOV, AND S. P. SUETIN Theorem 2.1 that D/(Ah)=V/A, with V defining Γ. Since Λ was an arbitrary convergent subsequence, we conclude that actually Dn V · ∞ lim Hn = lim = = G ( , ). n n Ahn A This establishes Lemma 3.6. For the polynomials Dn in (3.1) we have the representation + Dn(z)=Vn(z)hn(z), such that p −2 Vn(z)= (z − vk,n) → V (z),n→∞, k=1 + p−2 + and hn(z)=z + ... satisfies hn − hn → 0 as n →∞. Observe also that from our Conjecture 3.3 it follows that all zeros of V are simple. Since Hn appears multiplied by n in (3.4), we need to estimate the rate of convergence of Hn to G . Together with the Green function G it is convenient to consider also z Vn(t) Gn(z,∞):= dt. a1 A(t) The following lemma is an elementary observation: Lemma 3.7. We have ∞ ∞ Hn(z)=Gn(z, )(1+δh,n(z)+ε1(z)) = G (z, )(1+δh,n(z)+δV,n(z)+ε2(z)) , where + p−2 + hn(z) − hn(z) βk,n hn(zk,n) (3.5) δh,n(z):= = , with βk,n := , 2hn(z) z − zk,n 2h (zk,n) k=1 n p−2 Δvk,n (3.6) δV,n(z):= , with Δvk,n := vk,n − vk, z − vk k=1 O 2 O 2 →∞ εj = (δh,n(z)) + (δV,n(z)),n . √ Proof. Applying the identity 1+ξ =1+ξ/2+O(ξ2), ξ →∞,weget + Vn(z)hn(z) ∞ O 2 Hn(z)= = Gn(z, ) 1+δh,n(z)+ (δh,n(z)) . A(z)hn(z) By Assumption 3.3, zeros of hn are all simple for n large enough. Hence, using the + partial fraction decomposition for (hn − hn)/hn we obtain the second identity in (3.5). · ∞ Finally, differentiating Gn( , ) with respect to its parameters vk,n,weobtain · ∞ · ∞ O 2 that Gn( , )=G ( , ) 1+δV,n(z)+ (δV,n(z)) , with δV,n given in (3.6). In order to find the asymptotics for δh = δh,n we need the following result: Lemma 3.8. At any zero zk,n of hn we have 2 − (3.7) n(n +1)Dn = AhnDn (Ahn + Bhn)Dn . JOHN NUTTALL’S WORK 25 YEARS LATER 173 Proof. Differentiating (3.1) and evaluating the result at z = zk,n we get − 2 − 2 (3.8) Ahn + Bhn + N Dn Rn N DnRn =0, where 1 1 (3.9) N := n(n +1)=n + + O ,n→∞. 2 n Also from (3.1), for z = zk,n, 2 (3.10) AhnRn + N DnRn =0. (3.8)–(3.10) give us a homogeneous linear system on (Rn(z),Rn(z)) with a non- trivial solution, since by the uniqueness theorem, at a regular point of (3.1), both Rn and Rn cannot vanish simultaneously. Hence, the determinant of this system is zero, which yields the assertion. + Using that Dn = Vnhn we obtain from (3.7), 2 2+2 + + − N Vn hn = AhnhnVn + hn(AVnhn Ahn + Bhn)forz = zk,n, and since by Assumption 3.3, hn(zk,n) =0, (3.11) 2 +h A +h − h Ah V − Ah + Bh N 2 n = 1+ n n + +h n n n n for z = z . n 2 − k,n hn Vn hn (hn) AV n As a consequence, we get the following lemma: Lemma 3.9. For βk,n defined in (3.5), 2 1 A(zk,n) O 1 A(zk,n) O O βk,n = 2 (1 + (δh,n)) = 2 (1 + (δh,n)+ (δV,n)) . 4N Vn(zk,n) 4N V (zk,n) Observe that the last identity is obtained applying also Lemma 3.7. Next, we use the possibility of the analytic continuation in (3.4) in order to derive the asymptotic identities on the unknown parameters. Lemma 3.10. Let γ be a cycle (simple closed curve) in C \ en enclosing two points, say a1,a2 ∈A, in such a way that the rest of points from en are exterior to γ.Then O (3.12) N Hn(t) dt = T (γ,f)(1 + (1/n)), γ with sin πα T (γ,f):=± log 1 +2πim, m ∈ Z, sin πα2 where the sign is uniquely determined by the branch of the square root and the orientation of the contour γ chosen. Proof. By Theorem 3.4, for any aj ∈Athere exists a progressive path γj from O to O that is a closed Jordan curve separating aj from other points of en; assume γj positively oriented with respect to aj . Observe that both analytic germs f and Rn = Qnf − Pn in O allow for the − analytic continuations along any such a path. Denote by fγj and Rn,γj = Qnfγj Pn the values of these functions that we obtain after the analytic continuations of f 174 A. MART´INEZ-FINKELSHTEIN, E. A. RAKHMANOV, AND S. P. SUETIN and Rn, respectively, along γj .Ifwedenoteby−γj the negatively oriented contour γj ,thenforz ∈ γj ∩ O, ±2πiαj f±γj (z)=f(z)e . Consider for instance paths γ1 and −γ2. By the definition of the residue, Rn = Qnf − Pn,wehave − − Rn,−γ2 Rn f−γ2 f (3.13) − (z)= − (z). Rn,γ1 Rn fγ1 f But −2πiα −πiα f− − f e 2 − 1 e 2 sin πα f− sin πα (3.14) γ2 (z)= = − 2 = − γ2 (z) 2 . − 2πiα1 − πiα1 fγ1 f e 1 e sin πα1 fγ1 sin πα1 On the other hand, during the analytic continuation the residue of the Pad´e approx- imant picks up a dominant term; thus, Rn is geometrically small in O in comparison with Rn,i, i =1, 2 (see (3.4)), and from Theorem 3.4 we have for z ∈ O, − Rn,−γ2 Rn Rn,2 − (z)= (z)(1 + o(1)) Rn,γ1 Rn Rn,1 f− − γ2 (3.15) = (z)exp (n +1/2) Hn(t)dt (1 + o(1)) fγ1 γ1−γ2 f− − γ2 = (z)exp N Hn(t)dt (1 + o(1)) fγ1 γ1−γ2 (observe that the negative sign comes from the fact that the term A−1/4 is multiplied by ±i after its analytic continuation; orientations of γ1 and −γ2 are opposite, so after division we gain the −1 factor). Identities (3.13)–(3.15) yield the assertion with γ = γ1 − γ2 or any any cycle homotopic to it in C \ en. In order to extend the theorem to an arbitrary cycle γ in C \ en, we observe that if during the homotopic + deformation of the contour we cross a pair of adjacent zeros of hn and hn (see Lemma 3.6), both Rn,γ1 and Rn,−γ2 gain a change of sign, so that (3.12) remains valid. We introduce the Riemann surface R defined by the equation w2 = A(z)V (z). It is a hyperelliptic Riemann surface that can be considered as a two-sheeted cov- ering of C, R = {z =(z,w) ∈ C2}, with two sheets, R(1) and R(2), cut along Stahl’s compact Γ and glued together in the standard way. From Assumption 3.3 and using the Riemann-Hurwitz formula we easily see that the genus of R is p − 2. The canonical projection π : R→C is given by π(z)=z for z =(z,w) ∈ C2.We denote z(j) = π−1(z) ∩R(j), j =1, 2, and we convene that sheet R(1) over C \ Γ is specified by the condition w/z2 → 1asz → ∞(1) ∈R(1). In this way, function w =(AV )1/2 is single-valued on R. Unless specified otherwise, we identify the first sheet R(1) with the domain C \ Γ=π(R(1)). We construct analogously the Rie- 2 mann surface Rn defined by the equation w = A(z)Vn(z). Again, by Assumption 3.3, the genus of Rn is p − 2forn large enough. Note that a homology basis of cycles of R and Rn can be constructed from an integer combination of cycles γij = γ(ai,aj )=γi − γj considered in the proof of Lemma 3.10. Thus, (3.12) is valid for any cycle γ on R with the right hand side T (γ,f) depending on the representation of γ in terms of the basis of cycles γij.We JOHN NUTTALL’S WORK 25 YEARS LATER 175 select a homology basis of cycles Γj of R, j =1, 2,...,2p − 4, in such a way that, in the standard terminology, Γj are the a-cycles of R when j =1,...,p− 2, while for j = p − 1,...,2p − 4 they form the b-cycles. Next, we introduce a notation for some special functions and meromorphic differentials on R. Function V (t) 1 (z,t):= A(t) t − z can be regarded as meromorphic on R in both variables; dωk(t)=(vk, t) dt, k =1,...,p− 2, is a basis of holomorphic differentials on R, and correspondingly, z uk(z):= dωk,k=1,...,p− 2, a1 form a basis of integrals of the first kind (these are multivalued and analytic func- tions on R having a constant increment along any cycle). Additionally, A(z) ζ (3.16) θ(z, ζ):= (z,t)dt V (z) a1 can be also considered as an analytic function on R in both variables (multivalued in ζ). Lemma 3.7 and Lemma 3.9 render that with an appropriate choice of zk,n = −1 (1) (2) ∈ C \ π (zk,n), that means, either zk,n or zk,n,forz Γ, p−2 p−2 1 (3.17) NH (z)=NG(z,∞)+ d u (z)+ θ(z ,z)+O(1/n), n k,n k 2 k,n k=1 k=1 where dk,n := N(vk,n − vk)=NΔvk,n, and as usual, we identify C \ Γ with the first sheet of R. Let us work out the system of equations on the unknown parameters. Given a closed contour (cycle) γ on R,wedenoteby % % % A(z) V (t) dt ∈R\ Θ(z; γ):=Δγ θ(z, ζ)% = − , z γ, ζ∈γ V (z) γ A(t) t z the period of θ(z, ·)alongγ. A direct verification shows that Θ(·; γ) can be ana- lytically continued on R as an integral of the first kind, so that for suitably chosen ck(γ) ∈ C, p−2 (3.18) Θ(z; γ)= ck(γ)uk(z). k=1 By (3.12), O − (3.19) N Hn(t) dt = T (Γj,f)(1 + (1/n)) mod 2πi, j =1, 2,...,2p 4, Γj 176 A. MART´INEZ-FINKELSHTEIN, E. A. RAKHMANOV, AND S. P. SUETIN and in view of (3.17), equation (3.19) may be written as (3.20) p−2 p−2 1 N G(t, ∞) dt=T (Γ ,f)− d dω − Θ(z , Γ )+O(1/n)mod2πi, j k,n k 2 k,n j Γj k=1 Γj k=1 with j =1, 2,...,2p − 4. This is a system of 2p − 4 equations on 2p − 4 unknowns d1,n,...,dp−2,n, z1,n,...,zp−2,n, that may be equivalently written in any basis Γj . From the general theory of Riemann surfaces it follows that matrix p−2 dωk Γ j j,k=1 is invertible. Then, first p − 2 equations in (3.20) may be explicitly solved for dk,n. Substitution of those dk,n’s in the remaining equations and the use of (3.18) reduces the situation to the standard Jacobi inversion problem, which as it is well known, is uniquely solvable for any non-special divisor5. Hence, system (3.20) is uniquely solvable for any right hand side. Remark 3.11. Under Assumption 3.3, Δvk,n = O(1/n) and all the remainders in (3.20) are O(1/n), which is the accuracy for determining dk,n by these equations. Now we can simplify the asymptotic formula (3.4) from Theorem 3.4. Since 3/4 −1/4 1/2 −1/4 O hn Dn = hn V (1 + (1/n)), we get 1/2 (fhn) − R (z)=C (z) e NHn(z) (1 + O(1/n)) , n n,1 (AV )1/4 and Hn can be replaced by Hn, the leading term in its asymptotic formula (3.17): p−2 p−2 1 1 (3.21) H (z)=G(z,∞)+ d u (z)+ θ(z ,z). n N k,n k 2N k,n k=1 k=1 Finally, using the analytic continuation of Rn along a progressive path around an a ∈A(if we take a = a1,thenHn just changes sign during this analytic continuation) and solving the system $ Qnf − Pn = Rn Qnf1 − Pn = Rn,1 for Qn we obtain Qn =(Rn,1 − Rn)/(f1 − f), from where the exterior asymptotics has the form 1/2 hn Q (z)=C (z) eNHn(z) (1 + O(z)) . n n,1 f 1/2(AV )1/4 We summarize our findings in the following theorem: 5 In the situation when the divisor is special, we have deg hn JOHN NUTTALL’S WORK 25 YEARS LATER 177 Theorem 3.12. For a normal index n ∈ N let N = n(n +1) and Hn be as defined in (3.21), with the coefficients dk,n and zk,n determined by equations (3.20). Then, with an appropriate normalization, 1/2 (fhn) − H R (z)=C (z) e N n(z) (1 + O(1/n)) , n n,1 (AV )1/4 and 1/2 hn H (3.22) Q (z)=C (z) eN n(z) (1 + O(1/n)) , n n,2 f 1/2(AV )1/4 for z on a compact subsets of C \ Γ. 4. Case of p =3: Chebotarev compact and the Riemann surface In the rest of the paper we concentrate on the particular case studied in [30], when p =3,a1, a2 and a3 are 3 non-collinear points on the complex plane C,andΓ is the Chebotarev compact, i.e. the set of minimal capacity containing these points. Recall (see Theorem 2.1) that there exists a point v in the convex hull of A, called the center of the Chebotarev compact, such that with V (z) 1/2 A(z)=(z − a )(z − a )(z − a ),V(z)=z − v, and T (z)= , 1 2 3 A(z) where the branch of T in C \ Γ is specified by limz→∞ zT(z) = 1, it is determined uniquely by the set of equations v v Re T (t) dt =Re T (t) dt =0. a1 a2 Furthermore, Γ=Γ1 ∪ Γ2 ∪ Γ3, with $ ) z Γj := z ∈ C :Re T (t) dt =0 , aj the arc of Γ joining aj with v, j =1, 2, 3. We introduce also the orthogonal trajectories z Γ⊥ := z ∈ C :Im T (t) dt =0 , v which consist of 3 unbounded rays emanating from v,aswellas $ ) z ⊥ ∈ C γj := z :Im T (t) dt =0 ,j=1, 2, 3; aj ⊥ each γj is an unbounded ray emanating from aj , see Fig. 2. ⊥ ∪ ∪ ⊥ C\ Contour γ1 Γ γ2 splits Γ into two simply connected domains. We denote by D+ the domain containing a3 on its boundary, and D− the complementary one. On the three subarcs of Γ we fix the orientation “from aj to v”, while on the arcs of Γ⊥ we choose the orientation “from v to infinity”. This induces the left (“+”) and right (“−”) sides and boundary values. The equilibrium measure λ = λΓ on Γ has the form 1 dλ(z)= T−(z)dz πi 178 A. MART´INEZ-FINKELSHTEIN, E. A. RAKHMANOV, AND S. P. SUETIN ⊥ a3 γ3 ⊥ Γ2 Γ3 ⊥ γ1 ⊥ a1 Γ1 v Γ1 Γ2 ⊥ Γ3 a2 ⊥ γ2 Figure 2. ΓandΓ⊥. (compare with (2.5)). We denote also 1 v mj = λ(Γj)= T−(t)dt, j =1, 2, 3, πi aj so that m1 + m2 + m3 =1. Define in C \ Γ z (4.1) Φ(z)=exp T (t) dt , v normalized by the condition lim Φ(z)=1; z→v ∈ ⊥ z Γ1 observe that Φ coincides up to a multiplicative constant with exp (G(·, ∞)) intro- duced in (2.4). It is a conformal mapping of the exterior C \ Γ onto the exterior of the unit circle, such that (4.2) Φ(z)=cz+ O(1),z→∞, with 1/c coinciding, again up to a factor of absolute value 1, with the logarithmic capacity of Γ. Direct calculation allows to establish the following lemma: ◦ Lemma 4.1. For z ∈ Γ := Γ \{v, a1,a2,a3} and with the orientation shown on Figure 2, ∈ ◦ \{ } (4.3) Φ−(z)Φ+(z)=κj ,zΓj := Γj v, aj , with 2πi(m3−m2) −2πim2 2πim3 (4.4) κ1 = e ,κ2 = e ,κ3 = e , so that |κj | =1and κ2κ3 = κ1. As before, we consider the Riemann surface R defined by the equation w2 = A(z)V (z). Now it is an elliptic Riemann surface that can be considered as a two- sheeted covering of C, R = {z =(z,w) ∈ C2}, with two sheets, R(1) and R(2), cut along Γ and glued together in the standard way. The canonical projection π : R→C is given by π(z)=z for z =(z,w) ∈ C2. As in Section 3, we denote z(j) = π−1(z)∩R(j), j =1, 2, and we convene that sheet R(1) over C\Γ is specified JOHN NUTTALL’S WORK 25 YEARS LATER 179 a3 b-cycle a1 v a-cycle a2 R(1) a3 a1 v a2 R(2) Figure 3. Cycles on R. by the condition w/z2 → 1asz → ∞(1) ∈R(1). In this way, function w =(AV )1/2 is single-valued on R, with $ T (z)/V (z), for z = z(1) ∈R(1), w(z)= −T (z)/V (z), for z = z(2) ∈R(2). Again, we identify the first sheet R(1) with the domain C \ Γ=π(R(1)). We also (j) −1 (j) denote D± = π (D±) ∩R , j =1, 2. We define the canonical homology basis of cycles as in Figure 3: the a-cycle encloses v and a1, while the b-cycle goes around v and a3. Both are oriented as indicated in the figure, so that at their unique intersection point on R their tangent vectors form a right pair. The normal form of the Riemann surface R is the polygon (rectangle) R+ with sides aba−1b−1 (see Figure 4)6. We introduce also some notation, slightly different from that used in Section 3, related to differentials on R and their periods. For integer k,denote zkdz (4.5) dν (z)= . k w(z) Then dν0 is, up to a constant multiple, the only holomorphic differential (abelian differential of the first kind) on R. ∈R For two points r1, r2 we denote by Ωr1,r2 the normalized differential of the third kind, such that it has only simple poles at r1, with residue +1, and at r2, 6One of the authors of [2] kindly pointed out to us that a similar figure is contained in the cited paper. 180 A. MART´INEZ-FINKELSHTEIN, E. A. RAKHMANOV, AND S. P. SUETIN v Γ3+ a3 Γ3− v Γ D(1) 1+ Γ2− + (2) D− a1 a1 (2) (1) ∞ a2 ∞ (1) D− Γ2+ D(2) Γ1− + v a3 v Figure 4. Polygon R+. Shaded domain corresponds to the first R(1) −1 ⊥ ∪ ⊥ sheet , and the dotted line corresponds to π (γ1 γ2 ). with residue −1, and with a vanishing b-period. In particular, if z∗ =(z∗,w∗) ∈R, 1 w(z)+w∗ dΩ ∗ ∞(1) = + z + δ dz, z , z − ∗ 2w ( ) z z (4.6) ∗ − 1 w(z)+w dz δ = − ∗ + z dz . b w(z) z z b w(z) With the orientation of Γj specified in Figure 2 we define k k − 1 t dt ∈ N ∪{ } (4.7) Mj := ,k 0 ,j=1, 2, 3, 2πi Γj w+(t) so that − k − k ∈ N ∪{ } dνk = 4πiM1 , dνk = 4πiM3 ,k 0 . a b 0 In particular, Mj =0,j =1, 2, 3, and Im(τ) < 0, with 0 M1 (4.8) τ := 0 . M3 Direct computation using the Cauchy integral formula shows that 1 S M 0 + M 0 + M 0 =0,M1 + M 1 + M 1 = ,M2 + M 2 + M 2 = , 1 2 3 1 2 3 2 1 2 3 2 whereweusethenotation v + a + a + a (4.9) S := 1 2 3 . 2 ∗ We reserve the notation dν0 for the normalized differential of the first kind, whose b-period is equal 2πi: ∗ − 1 dz − 1 (4.10) dν0 (z)= 0 = 0 dν0. 2M3 w(z) 2M3 ∗ Observe that the a-period of ν0 is 2πiτ. JOHN NUTTALL’S WORK 25 YEARS LATER 181 5. Riemann and Hilbert, or the non-linear steepest descent analysis Now we are ready to return to the Pad´e approximants of the function α1 α2 α3 f(z)=(z − a1) (z − a2) (z − a3) , with α1,α2,α3 ∈ R \ Z, such that α1 + α2 + α3 = 0. For the sake of simplicity of the analysis we assume additionally that αj > −1, j =1, 2, 3. As before, we specify the branch in C \ Γbyf(∞) = 1 and agree in denoting by f 1/2 the branch of the square root in C \ Γgivenbyf 1/2(∞)=1. We need to introduce an additional piece of notation: for j =1, 2, 3, let −iπαj n (5.1) τj = e ,tj =2i sin(παj),sn,j = tj κj , with κj defined in (4.4). Observe that τ1τ2τ3 =1,and −1 −1 (5.2) τj+1tj−1 + tj + tj+1τj−1 = τj−1tj+1 + tj + tj−1τj+1 =0,j=1, 2, 3, where the subindices are taken mod 3. Let us collapse the contour of integration in (2.3) onto Γ; as a consequence, function f induces on Γ the weight ρ, − tj −2 − ∈ \{ } ρ(z)=f−(z) f+(z)= f+(z)=(τj 1)f+(z),zΓj v, aj ,j=1, 2, 3, τj so that the orthogonality condition (2.3) can be rewritten as k (5.3) t Qn(t)ρ(t) dt =0,k=0, 1,...,n− 1. Γ By our assumption that αj > −1, the weight is integrable on Γ, and the regularity of f at infinity implies that lim ρ(z) + lim ρ(z) + lim ρ(z)=0. z→v z→v z→v z∈Γ1 z∈Γ2 z∈Γ3 Standard arguments show that there is an integral formula for the residue R : n 1 Qn(t)ρ(t) ∈ C \ Rn(z)= − dt, z Γ. 2πi Γ t z This allows us to formulate the Riemann-Hilbert problem for Qn and Rn.Let σ denote the third Pauli matrix, 3 10 σ = , 3 0 −1 and for any scalar a we use the notation a 0 aσ3 = . 0 a−1 We seek the matrix-valued and analytic function Y = Y (·; n):C \ Γ → C2×2,such that: ◦ (RH-Y1) It has continuous boundary values Y± on both sides of Γ , and with the specified orientation of Γ, 1 ρ(z) ◦ Y (z)=Y−(z) ,z∈ Γ . + 01 (RH-Y2) Y (z)=(I + O(1/z)) znσ3 ,asz →∞. 182 A. MART´INEZ-FINKELSHTEIN, E. A. RAKHMANOV, AND S. P. SUETIN (RH-Y3) As z → aj , z ∈ C \ Γ, j =1, 2, 3, ⎧ | − |αj ⎪ O 1 z aj ⎨ αj , if αj < 0, 1 |z − aj | Y (z)= ⎪ 11 ⎩ O , if α > 0. 11 j (RH-Y4) As z → v, z ∈ C \ Γ, 1log|z − v| Y (z)=O . 1log|z − v| From the fundamental work of Fokas, Its and Kitaev [14] it follows that Theorem 5.1. The matrix valued function Y (z) given by Qn(z) Rn(z) Y (z)= − 2 − 2 2πiγn−1Qn−1(z) 2πiγn−1Rn−1(z) is the unique solution of (RH-Y1)–(RH-Y4), where Qn is the monic polynomial of degree n satisfying (5.3) and γn is the leading coefficient of the corresponding orthonormal polynomial. This result is complemented with the non-linear steepest descent method of Deift and Zhou [6, 9, 10, 11, 12]: we need to perform a number of explicit and invertible transformations of (RH-Y1)–(RH-Y4) in order to reach a boundary value problem with jumps asymptotically close to the identity and a regular behavior at infinity. Two of the main ingredients of this analysis are the outer (global) parametrix and the local model at the Chebotarev center, that we explain next. 5.1. Global parametrix. For n ∈ N we need to find an analytic matrix- 2×2 valued function Nn = N : C \ Γ → C , such that ◦ (RH-N1) It has continuous boundary values N± on both sides of Γ , and with the orientation “from a to v”ofΓ, j 0 sn,j ∈ ◦ (5.4) N+(z)=N−(z) ,zΓj . −1/sn,j 0 (RH-N2) N(z)=I + O(1/z)), as z →∞. (RH-N3) As z → aj , z ∈ C \ Γ, j =1, 2, 3, −1/4 N(z)=O(|z − aj | ). As z → v, z ∈ C \ Γ, N(z)=O(|z − v|−1/4). Constants sn,j were defined in (5.1); hence, the dependence on n resides only in the boundary condition (5.4). On the compact subsets of C \ Γ this problem is asymptotically close to the boundary value problem for the following matrix, − (5.5) T (z):=cnσ3 Y (z)Φ nσ3 (z)f σ3/2(z), with c defined in (4.2). Hence, we can expect that away from the Chebotarev compact Γ the solution N of (RH-N1)–(RH-N3) models the behavior of T for n large enough. JOHN NUTTALL’S WORK 25 YEARS LATER 183 We build N in the following form, σ −σ (5.6) N(z):=F (∞) 3 N(z)F (z) 3 , using two “ingredients” described in detail below: a scalar function F , which plays the role of a Szeg˝o function with piece-wise constant boundary values, and a matrix- valued function N =(Nij), which will be defined in terms of abelian integrals on R.BothF and N depend on n, but in this section we omit this dependence from the notation, keeping it in mind. With the notation (4.5) consider the equation zn 1 sn,2 1 sn,2 dν0 = − 1+ log dν0 − log dν0, ∞(1) 2πi sn,1 a 2πi sn,3 b or equivalently, zn ∗ − 1 sn,2 − sn,2 (5.7) dν0 = 2πiτ 1+ log log , ∞(1) 2πi sn,1 sn,3 ∗ where ν0 is the normalized differential of the first kind (4.10), and the path of integration lies entirely in the rectangle R+. Among all possible choices of the branch of the logarithm, there is at most one value of log(sn,2/sn,1)andatmost + one value of log(sn,2/sn,3) such that this equation has a solution in R; this solution zn =(zn,wn) is obviously unique. Remark 5.2. If zn falls on one of the cycles Γj we consider it slightly deformed so that the same argument applies. A truly special situation occurs when eventually z = ∞(1) or z = ∞(2). The first case happens when n n M 0 s M 0 s (5.8) 1 log n,2 + 3 log n,2 ≡ 0modZ. 2πi sn,1 2πi sn,3 The consequences of this degeneration are discussed below, see Remark 5.8. With this choice of the branch of the value of log(sn,2/sn,1) we define two parameters, β1 and β2, as follows: (5.9) β := log s =log(t )+2πin(m − m ). 1 n,1 1 3 2 sn,2 (5.10) β2 :=πi +log . sn,1 Obviously, (5.9) defines β1 up to an additive constant which is an integer multiple of 2πi. With these two complex constants fixed, we build a complex-valued function F on C \ Γ, holomorphic, uniformly bounded and non-vanishing in C \ Γ, and such that ⎧ β1 ⎨⎪e = sn,1,z∈ Γ1 \{v, a1}, β +β (5.11) F+(z)F−(z)= e 1 2 = −s ,z∈ Γ \{v, a }, ⎩⎪ n,2 2 2 β1+β3 β3 e = sn,1e ,z∈ Γ3 \{v, a3}. Constant β3 is not arbitrary: (5.12) β3 =(1+τ) β2, with τ from (4.8). We take F of the form F (z)=exp(Λ(z)), and give two equivalent expressions for Λ. 184 A. MART´INEZ-FINKELSHTEIN, E. A. RAKHMANOV, AND S. P. SUETIN First, Λ can be built in terms of the holomorphic differential ν0 on R: z(1) z(1) β1 β1 dt (1) (1) (5.13) Λ(z)= +Ξ dν0 = +Ξ , z ∈R ,z∈ C \ Γ, 2 a1 2 a1 w(t) where 2 − 2 −S 1 − 1 (5.14) Ξ = β2 M1 τM3 M1 τM3 , k S with Mj introduced in (4.7) and in (4.9). The path of integration in (5.13) lies entirely in R(1), except for its initial point. Observe that Λ in (5.13) is a holomorphic function in C \ Γ. Alternatively, define the functions w(z) dt ∈ C \ (5.15) lj (z):= − ,z Γ,j=1, 2, 3, 2πi Γj w+(t)(t z) where we integrate in the direction “from aj to v”, and let β (5.16) Λ(z)= 1 − β l (z) − τl (z) − 1/2 ,z∈ C \ Γ. 2 2 1 3 Lemma 5.3. With Λ given either by (5.13)–(5.14) or by (5.16), function F (z)= Fn(z)=exp(Λ(z)) is holomorphic, uniformly bounded and non-vanishing in C \ Γ, with β (5.17) F (∞)=exp 1 − β M 1 − τM1 − 1/2 . 2 2 1 3 Moreover, F has continuous boundary values at Γ◦ that satisfy (5.11)–(5.12). In consequence, formulas (5.13)–(5.14) and (5.16) define the same function in C \ Γ. Remark 5.4. Recall that β2 was defined uniquely as a function of n, but β1 is determined mod (2πi). From (5.13)–(5.14) or (5.16) it follows that for each n, function F is determined uniquely up to a change of sign. Now we define in C \ Γ the analytic matrix valued function N entry-wise in terms of meromorphic differentials on R as follows (see also [20]). The meromorphic differential ⎛ ⎞ 1 (AV )(z) 1 1 1 3 1 1 dη∗(z)=− dz − dν∗ = ⎝− − + ⎠ dz 4 (AV )(z) 2 0 4 z − v z − a M 0 w(z) j=0 j 3 has only simple poles on R: at the zeros of AV with residues −1/2, and at ∞(1), ∞(2), both with residues +1; additionally, its b-period is zero. With zn =(zn,wn) solving (5.7) we consider the meromorphic differential ∗ (1) (5.18) ηzn = η +Ωzn,∞ , or more explicitly, − (AV ) (z) 1 wn z δ1 (5.19) dηzn = + + + + dz, 4(AV )(z) 2(z − zn) 2w(z)(z − zn) 2w(z) w(z) where δ1 is uniquely determined by the condition that the b-period of ηzn is zero. Obviously, it has the only poles, all simple, at the zeros of AV with residues −1/2, (2) and at zn and ∞ , both with residues +1. JOHN NUTTALL’S WORK 25 YEARS LATER 185 Lemma 5.5. With the conditions above, − (5.20) dηzn = β1 + β3 log(sn,3)mod(2πi), a with β1 and β3 given in (5.9) and (5.12), respectively. Proof. Direct calculation shows that dη∗ = πi(1 − τ) , a and from the Riemann identities it follows that zn zn 1 ∗ (5.21) Ω (1) = dν = − dν . zn,∞ 0 0 0 a 2M3 ∞(1) ∞(1) It remains to use (5.7) and the definition of β1 and β2 above. Remark 5.6. The uniqueness of zn satisfying (5.20) can be easily established: ∈R − R for any other r , νzn νr is a meromorphic differential in whose only poles (both simple) are at zn (with# residue 1) and r (with residue −1), and with z − periods multiple of 2πi,sothatexp( d(νzn νr)) has a single pole at r,whichis impossible. Let z(1) z(2) (5.22) u1(z)=exp dηzn ,u2(z)=exp dηzn , ∞(1) ∞(1) (j) −1 (j) (1) with z = π (z) ∩R .Foru1, the path of integration lies entirely in R , (1) (2) while for u2 it goes from D± into D∓ , crossing Γ2 once, see Figure 5. v a3 v z(1) D(1) Γ2− + (2) D− u1 a1 a1 (2) (1) ∞ a2 ∞ (1) D− (2) u2 z Γ2+ D(2) + v a3 v Figure 5. Paths of integration for functions uj defined in (5.22). Lemma . C \ ⊥ ∪ ∪ ⊥ 5.7 Functions uj are holomorphic in (γ1 Γ γ2 ) (see Figure 2), ⊥ ∪ ◦ ∪ ⊥ and have continuous boundary values on γ1 Γ γ2 such that (5.23) ⎧ ⎧ ⎪ ∈ ⊥ ∪ ⊥ ⊥ ⊥ ⎪(u1)−(z),z γ1 γ2 , ⎪− ∈ ∪ ⎨⎪ ⎨⎪ (u2)−(z),z γ1 γ2 , ∈ ◦ ∪ ◦ ◦ ◦ (u2)−(z),z Γ1 Γ2, ∈ ∪ (u1)+(z)= (u2)+(z)= (u1)−(z),z Γ1 Γ2, ⎪ β1+β3 ⎪ ⎪ e ◦ ⎪ sn,3 ◦ ⎩⎪− ∈ ⎩ − ∈ (u2)−(z),z Γ3; β +β (u1) (z),z Γ3. sn,3 e 1 3 186 A. MART´INEZ-FINKELSHTEIN, E. A. RAKHMANOV, AND S. P. SUETIN −1/4 Moreover, as z → aj , z ∈ C \ Γ, j =1, 2, 3, uk(z)=O(|z − aj | ), while as z → v, z ∈ C \ Γ, u (z)=O(|z − v|−1/4). Additionally, k 1 1 u (z)=1+O ,u(z)=O ,z→∞. 1 z 2 z (1) Finally, if zn ∈R then u1 has a simple zero at z = zn and u2(zn) =0 ;otherwise, u2 has a simple zero at z = zn and u1(zn) =0 . With these two functions we define in C \ Γ $ ∈ + + u2(z), if z D+, (5.24) N11(z)=u1(z), N12(z)= −u2(z), if z ∈ D−. ⊥ ∪ ∪ ⊥ Recall that the simply connected domains D± are limited by γ1 Γ γ2 ,andD+ is the one containing a3 on its boundary. Remark 5.8. As we have pointed out before, it may happen that either zn = (1) (2) ∞ or zn = ∞ . In the first case, when condition (5.8) holds, we just have ∗ ηzn = η ,sothatN11 has a zero at infinity. This, as it follows from the asymptotic formulas below, will mean that the index n is not normal, see the expression of χ ∞(2) in (6.1). In the second case, ηzn = η∞(2) has a simple pole at with residue +2, which creates a double zero of N12 at infinity. Furthermore, consider a family of functions q on R of the form w(z)+w q(z)=a + b n − z ,a,b∈ C. z − zn (2) Each such a function has a simple pole at zn =(zn,wn)andat∞ . There is a unique combination of constants a, b, such that additionally (1) q(∞ ) = 0 and lim q(z)N12(z)=1. z→∞(2) Let q(j)(z)=q(z(j))bethevaluesofq on the j-th sheet. Then set (1) (2) N21(z)=q (z)N11(z), N22(z)=q (z)N12(z). This defines completely the matrix-valued function N =(Nij). Finally, we assem- bly N as in (5.6) using this matrix N and function F given by (5.13)–(5.14) or (5.16) with parameters (5.9)–(5.10). Direct verification shows that the following statement holds true: Proposition 5.9. Matrix N constructed above solves the RH problem (RH- N1)–(RH-N3). 5.2. Local parametrix. Matrix T defined in (5.5) has jumps that are asymp- totically close to the identity matrix for n large enough, as long as we stay away from Γ. However, this behavior fails in a neighborhood of A and the Chebotarev center v, where we need to perform a separate analysis in order to find an appro- priate model. Here we describe only the construction of the local parametrix P at z = v (around the branch points aj matrix P is built in the way described in detail in [19]). We take a small δ>0 and define Dδ := {z ∈ C : |z − v| <δ}, Bδ := {z ∈ C : |z − v| = δ}, assuming that Dδ ∩A= ∅, see Figure 6. JOHN NUTTALL’S WORK 25 YEARS LATER 187 ⊥ Γ2 ⑥ Γ3 ① ⑤ ⊥ Γ1 Γ1 v ② ④ ③ ⊥ Γ2 Γ3 Figure 6. Local parametrix. The local parametrix at z = v has the form 2/3 σ3 −nσ3 ⊥ (5.25) P(z):=E(z)Ψ n ϕ(z) b B Φ (z),z∈ Dδ \ (Γ ∪ Γ ). Here Φ is the conformal mapping defined in (4.1), and the rest of the ingredients are: ⊥ • Function ϕ, defined piece-wise in each sector: for z ∈ Dδ \ (Γ ∪ Γ ), ⎧ ⎪ 3 z 2/3 ⎨⎪ T (t)dt if z ∈ ④ ∪ ⑤, 2 (5.26) ϕ(z):= v ⎪ 3 z 2/3 ⎩⎪ − T (t)dt otherwise, 2 v where we take the main branch of the power function. Then ϕ is a con- formal mapping of Dδ onto a neighborhood of the origin, ϕ(v) = 0, and ⊥ Γ1 is mapped onto the positive semi axis. • Constants b1 and b2, determined up to a change of sign by b1b2 =1/t1,b1/b2 = −t2/t3, and $ b , if z ∈ ① ∪ ⑤ ∪ ⑥, b := 1 b2, if z ∈ ② ∪ ③ ∪ ④. • Matrices ⎧ ⎪ ∈ ① ∪ ② ∪ ③ ∪ ⑥ ⎪ I, if z , ⎪ ⎨⎪ 0 −t2 , if z ∈ ④, B := ⎪ 1/t2 0 ⎪ ⎪ 0 t3 ⎩⎪ , if z ∈ ⑤, −1/t3 0 and ⎧ n σ3 ∈ ① ∪ ⑥ ⎪(κ3 /b1) , if z , ⎨⎪ n σ3 ∈ ② ∪ ③ (κ2 /b2) , if z , M2 = M2(z,n):= ⎪ 0 b t ⎩⎪ 2 2 , if z ∈ z ∈ ④ ∪ ⑤. −1/(b2t2)0 188 A. MART´INEZ-FINKELSHTEIN, E. A. RAKHMANOV, AND S. P. SUETIN • Matrix valued functions in Dδ \ Γ1, √ eπi/6 −eπi/6 M (z):= π ϕσ3/4(z), 1 e−πi/3 e−πi/3 where we take the main branch of the root, and σ3/6 (5.27) E(z)=E(z,n):=N(z)M2(z)M1(z)n ,z∈ Dδ \ Γ, with N constructed in subsection 5.1, see (5.6). E(z) extends in fact as a holomorphic function to the whole Dδ. • The Airy parametrix, Ψ, defined as 2 Ai(ζ) Ai(ω ζ) − − Ψ(ζ):= ω σ3/4,ζ∈ ϕ 1(⑤ ∪ ⑥), Ai(ζ) ω2Ai(ω2ζ) 2 Ai(ζ) Ai(ω ζ) − 10 − Ψ(ζ):= ω σ3/4 ,ζ∈ ϕ 1(①), Ai(ζ) ω2Ai(ω2ζ) −11 2 Ai(ζ) −ω Ai(ωζ) − − Ψ(ζ):= ω σ3/4,ζ∈ ϕ 1(③ ∪ ④), Ai(ζ) −Ai(ωζ) 2 Ai(ζ) −ω Ai(ωζ) − 10 − Ψ(ζ):= ω σ3/4 ,ζ∈ ϕ 1(②), Ai(ζ) −Ai(ωζ) 11 where ω =exp(2πi/3), see e.g. [7, 8]. Theorem 5.10. Matrix-valued function P given by (5.25) solves the following boundary-value problem in Dδ: (RH-P1) It has continuous boundary values P± on both sides of all curves, and with the specified orientation, ◦ ⊥ P+(z)=P−(z) JP (z),z∈ (Γ ∪ Γ ) ∩ Dδ, where 0 sn,j ◦ (5.28) J = − , if z ∈ Γ ∩ D ; P −s 1 0 j δ n,j 10 ⊥ (5.29) = t − , if z ∈ Γ ∩ D . j Φ 2n(z)1 j δ tj−1tj+1 (RH-P2) P (z)=(I + O(1/n)) N(z),forz ∈ Bδ. (RH-P3) As z → v, z ∈ C \ (Γ ∪ Γ⊥), P (z)=O(1). 5.3. Asymptotic analysis. In the final transformation of the original prob- lem (RH-Y1)–(RH-Y4) we define a matrix valued function R in the form T (z)A−1(z), where, roughly speaking, A = N away from Γ and A = P in a neighborhood of A and v. The explicit formula for A in Dδ is given above, while it is built in terms of the Bessel functions in a neighborhood of the branch points aj , see [19] for details. The inverses of all these matrices exist, since the determinants of these matrices are equal to 1. The construction of N and P is such that 1 R (z)=R−(z) I + O ,n→∞, + n JOHN NUTTALL’S WORK 25 YEARS LATER 189 uniformly on a finite set of contours in C. Following the already standard reasoning we conclude that 1 (5.30) R(z)=I + O ,n→∞, n uniformly in C. The relation (5.30) is the main term in the asymptotics for R and it is enough to give the leading term in the asymptotics for Y . For instance, taking into account (5.5) and (5.30) we see that locally uniformly in C \ Γwehave − 1 − Y(z)=c nσ3 I + O N(z)f(z) σ3/2Φnσ3 (z), n where c was defined in (4.2). In particular, working out the expressions for Y11 and Y12 we get: Theorem 5.11. Locally uniformly in C \ Γ, (5.31) (1) Φ(z) n F (∞) z 1 1 Q (z)= exp dη 1+O + O , n 1/2 zn c f(z) (z)F (z) ∞(1) n n and 1/2 z(2) f(z) ∞ ± O 1 O 1 Rn(z)= n F ( )F (z) exp dηzn 1+ + . (cΦ(z)) ∞(1) n n The sign and the paths of integration are selected in accordance with the defi- nition of N1j in (5.24). We see in particular, that the spurious zero of Qn is asymptotically close to the unique zero of N11, which appears only when zn is on the first sheet. Otherwise, it gives us an extra interpolation condition (zero of Y12, close to the zero of N12). The Riemann-Hilbert analysis yields asymptotic formulas not only away from Γ but in the rest of the regions. For instance, close to Γ but still away from the branch points aj and the Chebotarev center v the asymptotic expression for Qn is a combination of two competing terms, which gives rise to zeros of Qn. For instance, by (5.5) and (5.30), for z in ① of the domain D, − 1 10 − nσ3 O σ3/2 nσ3 Y(z)=c I + N(z) τ1 f(z) Φ (z). 2n 1 n t1Φ (z) In particular, n 1/2 n τ1 −n 1 c Qn(z)f(z) = N11(z)Φ (z)+N12(z) Φ (z) 1+O t1 n n τ1 −n 1 + N21(z)Φ (z)+N22(z) Φ (z) O . t1 n For z ∈ Γ1 we can rewrite it as n Φ (z) N (z) N (z)− 1 Q (z)= + 11 + + 11 1+O . n c 1/2 1/2 n f(z)+ f(z)− 190 A. MART´INEZ-FINKELSHTEIN, E. A. RAKHMANOV, AND S. P. SUETIN Finally, in order to analyze the behavior at the Chebotarev center v, for in- stance, when z ∈ Dδ ∩ ① (see Figure 6), it is sufficient to obtain the expression for Y from − 1 10 − nσ3 O σ3/2 nσ3 Y(z)=c I + P (z) τ1 f(z) Φ (z), 2n 1 n t1Φ (z) where all the ingredients in the right hand side were given above. Since the formulas for Qn and Rn obtained this way are not easily simplified, we omit their explicit calculation here for the sake of brevity. 6. Wrapping up, or matching the asymptotic formulas and the Nuttall conjecture In [29] Nuttall conjectured the form of the leading term of asymptotics for Qn and Rn away from Γ in terms of a solution of a scalar boundary value problem. We show next that our results match the Nuttall conjecture. Letusdenote n Φ(z) N (z) − (6.1) χ(z):= 11 , R(z):=(cΦ(z)) n N (z)f(z)1/2, c f(z)1/2 12 so that χ has a pole of order n at infinity, and R has there a zero of order n +1 (unless the pathological situation of zn = ∞ occurs). By (5.31), 1 Q (z)=χ(z) 1+O ,z∈ C \ Γ. n n ∈ ◦ But for z Γj , n 1/2 Φ(z)± (f(z)χ(z)) = f(z) N ±(z); ± ± c 11 ◦ using that ρ(z)=tj f+(z)/τj = tj τjf−(z)onΓj , (5.4) and Lemma 4.1, we get 1/2 f(z)+ −n σ(z)χ+(z)=− w+(z) (c Φ(z)−) N12−(z)=(wR)−(z), τj 1/2 −n σ(z)χ−(z)=w+(z)τjf(z)− (c Φ(z)+) N12+(z)=(wR)+(z), ◦ where σ(z):=ρ(z)w+(z)onΓ. These two equations match the boundary value conditions in [30, formula (5.6)] (after replacing χ2 = χ and H = wR) that define uniquely the leading asymptotic terms for Qn and Rn, according to the conjecture of Nuttall. Let us finally compare the asymptotic formulas obtained in Sections 3 and 5, and given by Theorems 3.12 and 5.11. We introduce here the notation z dt z u(z)= = dν0. a1 w(t) a1 On one hand, observe that in the case p = 3, with the function θ(z, ζ) defined in (3.16) we have z wn dt θ(zn,z)= u(z)+wn − , V (zn) a1 w(t)(t zn) JOHN NUTTALL’S WORK 25 YEARS LATER 191 so that 1 NHn(z)=NG(z,∞)+dn u(z)+ θ(zn,z) 2 z z wn dt 1 tdt O 1 =n log Φ(z)+ − + + δ2u(z)+ , 2 a1 w(t)(t zn) 2 a1 w(t) n where v wn δ2 = dn − + . 2 2V (zn) On the other hand, up to a multiplicative constant, z(1) z(1) −1 − F (z) exp dηzn =exp d(ηzn Ξ ν0) . Recalling the expression for ηzn in (5.19), we get z(1) −1 F (z) exp dηzn = (1) (z − z )1/2 z w z dt 1 z tdt = n exp n + dt+δ u(z) , 1/4 − 3 (AV ) (z) 2 a1 w(t)(t zn) 2 a1 w(t) with an appropriate selection of the constant δ3. Comparing thus expressions (3.21)–(3.22), obtained by the WKB analysis, with (5.31) we see that they co- incide, up to the right determination of the constants δ2 and δ3 above. 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Strebel, Quadratic differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 5, Springer-Verlag, Berlin, 1984. MR86a:30072 [47] S. P. Suetin, On the uniform convergence of diagonal Pad´e approximants for hyperelliptic functions, Mat. Sb. 191 (2000), no. 9, 81–114. MR1805599 (2002e:30034) [48] , On the asymptotics of diagonal Pad´e approximants for hyperelliptic functions of genus g = 2, Uspekhi Mat. Nauk 58 (2003), no. 4(352), 159–160. MR2042915 (2005c:41025) Department of Statistics and Applied Mathematics University of Almer´ıa, Spain, and Instituto Carlos I de F´ısica Teorica´ y Computacional, Granada University, Spain E-mail address: [email protected] Department of Mathematics, University of South Florida E-mail address: [email protected] V. A. Steklov Mathematical Institute of the Russian Academy of Sciences, Moscow, Russia E-mail address: [email protected] Contemporary Mathematics Volume 578, 2012 http://dx.doi.org/10.1090/conm/578/11484 Orthogonal Polynomials and S-curves E. A. Rakhmanov Dedicated to Francisco (Paco) Marcell´an on the occasion of his 60th birthday. Abstract. This paper is devoted to a study of S-curves, that is systems of curves in the complex plane whose equilibrium potential in a harmonic external field satisfies a special symmetry property (S-property). Such curves have many applications. In particular, they play a fundamen- tal role in the theory of complex (non-hermitian) orthogonal polynomials. One of the main theorems on zero distribution of such polynomials asserts that the limit zero distribution is presented by an equilibrium measure of an S-curve associated with the problem if such a curve exists. These curves are also the starting point of the matrix Riemann–Hilbert approach to strong asymptotics. Other approaches to the problem of strong asymptotics (differential equations, Riemann surfaces) are also related to S-curves or may be interpreted this way. Existence problem S-curve in a given class of curves in presence of a nontrivial external field presents certain challenge. We formulate and prove a version of existence theorem for the case when both the set of singularities of the external field and the set of fixed points of a class of curves are small (in main case — finite). We also discuss various applications and connections of the theorem. 1. Complex orthogonal polynomials and S-curves 1.1. Introductory Example. There are two types of orthogonal polynomi- als. The more usual and older type of orthogonality is hermitian one with respect to a positive weight (measure). Associated polynomials present, in particular, basis in weighted Hilbert spaces and used for polynomial approximation. In the last few decades another type of orthogonal polynomials — the complex (non-hermitian) ones with analytic weights came to attention. They appear first of all as denomina- tors of Pad´e approximants and other kinds of “free poles rational approximations”. Such polynomials are related, for instance, to continued fractions and three terms recurrence relations. They satisfy certain model differential equations. Both types are also related to their particular boundary value problems and to their own equi- librium problems. Equilibrium problems related to complex orthogonal polynomials is the main topic of the paper. We begin with an example of orthogonality with varying weights of the real line. In this case the two types of orthogonality coincide and associated orthogonal polynomials have all the properties mentioned above. 1991 Mathematics Subject Classification. Primary 30E15; Secondary 33E30, 34L20. c 2012 American Mathematical Society 195 196 E. A. RAKHMANOV Let Φn(x) be a sequence of continuous real-valued functions on Γ = [−1, 1], f(x) > 0a.e.onΓ.Let −2nΦn(k) (1.1) fn(x)=e f(x),x∈ Γ and polynomials Q (x)=xn + ... are defined by orthogonality relations n k (1.2) Qn(x) x fn(x) dx =0,k=0, 1,...,n− 1. Γ The following theorem by A. A. Gonchar and the author was probably the first gen- eral result on zero distribution of orthogonal polynomials with varying (depending on the degree of the polynomial) weight; see [22]. Theorem 1.1. If Φn(x) → Φ(x) uniformly on Γ then 1 1 ∗ X (Q )= δ(ς)→λ n n n Qn(ς)=0 where λ = λφ is the equilibrium measure of Γ is the external field Φ. In terms of the (total) energy 1 (1.3) E (μ)= log dμ(x) dμ(y)+2 Φ(x) dμ(x), ϕ |x − y| the equilibrium measure is defined by the minimization property in class M(Γ) of all unit positive Borel measures M on Γ (1.4) Eϕ(λ)= min Eϕ(μ). M∈M(Γ) The actual theorem in [22] was more general. It was formulated for Γ = R (which requires certain conditions on Φ(x)asx →±∞), convergence of Φn(x)toΦ(x)was of a weaker type. It was also noticed that Γ may be replaced# by a system of curves in k | | complex plane if standard hermitian orthogonality Γ Qn(x)x f(x) dx = 0 in (1.2) is assumed. Finally, it is possible to consider complex valued f(x): the theorem is still valid under some assumption on |f(x)| and arg f(x). Situation changes if Φ(x)=ϕ(x)+iϕ+(x) is complex valued. In this case we have first of all assume that f(x), Φn(x) and, therefore, Φ(x) are analytic in a domain Ω ⊃ (−1, 1) (otherwise the problem is illposed). Then Γ = [−1, 1] in (1.2) may be replaced by any rectifiable curve Γ in Ω connecting points −1, 1 (existence of integral near those points is assumed). Then orthogonality relation (1.2) are preserved; orthogonality become non-hermitian (complex). It turns out that assertion of Theorem 1.1 remains valid (with Φ replaced by ϕ = Re Φ) if there exists a curve S in class T of curves Γ ⊂ Ω connecting −1, 1 with a special symmetry property (S-property) for its total potential 1 V μ(z)+ϕ(z)= log dμ(x)+ϕ(x) |z − x| with external field ϕ. This is a simple particular case of a general theorem (Theorem 1.3) below. We note that the equilibrium measure λ = λϕ,Γ for a compact Γ in the external field ϕ defined by (1.3) is equivalently defined by the following relation in terms of total potential V λ + ϕ (x)=w, x ∈ supp λ (1.5) ≥ w, x ∈ Γ ORTHOGONAL POLYNOMIALS AND S-CURVES 197 (Equation (1.5) uniquely defines pair of measure λ ∈M(Γ) and constant w = wϕ,Γ — equilibrium constant.) Definition 1.2. Let S be a compact in C and ϕ be a harmonic function in a neighborhood of S.WesaythatS has an S-property relative to external field ϕ if there exist a set e of zero capacity such that for any ζ ∈ S ∼ e there exist a neighborhood D = D(ζ) for which supp(λ) ∩ D is an analytic arc and, furthermore, we have ∂ ∂ (1.6) V λ + ϕ (ζ)= V λ + ϕ (ζ) ζ ∈ supp(λ) ∼ e ∂n1 ∂n2 where λ = λϕ,S is the equilibrium measure for S in ϕ and n1,n2 are two oppositely directed normals to S at ζ ∈ S (we can actually admit that ϕ has a small singular set included in e). So, one of central for this paper concepts is defined by the pair of conditions (1.5)–(1.6) (with Γ = S). In particular, it follows by (1.5) that distribution of a positive charge presented by λ is in the state of equilibrium on the fixed conductor S. On the other hand, S-property of compact S in (1.6) in electrostatic terms means that forces acting on element of charge at ζ from two sides of S are equal. So, the equilibrium distribution λ of an S-curve will remain in equilibrium if we remove the condition that the charge belongs to S and make the whole plane a conductor (except for a few insulating points — endpoints of some of arcs in support of λ). Thus, λ presents a distribution of charge which is in equilibrium in conducting domain; such an equilibrium is unstable. In terms of energy (1.4) the S-property (1.6) of equilibrium measure is equiva- lent to the fact the λ is a a critical point of weighted energy functional with respect to local variations. We will go into further details in Section 4 below; technically speaking, variations of equilibrium energy is one of two fundamental components in the proof of the main result of the paper. This was, in short, electrostatic characterization of the limit zero distributions of complex orthogonal polynomials. 1.2. General theorem on zero distribution of complex orthogonal polynomials. Let Ω be a domain in C, S be a compact in Ω. Further, let Φn(z) ∈ H(Ω) and Φn(z) → Φ(z) uniformly on compacts in Ω as n →∞. Finally, n let f ∈ H(Ω ∼ S) and polynomials Qn(z)=z + ··· are defined by orthogonality − relations with weights f = fe 2nΦn n k (1.7) Qn(z)z fn(z)dz =0,k=0, 1,...,n− 1; S where integration goes over the boundary of C ∼ S (if such integral exist, otherwise integration goes over an equivalent cycle in C ∼ S). The following is a complex version of Theorem 1.1 (see [24]). Theorem 1.3. If S has S-property in ϕ =ReΦ(z) and complement to the 1 X →∗ support of equilibrium measure λ = λϕ,S is connected then n (Qn) λ. Thelastassertionisequivalenttoconvergence % % 1 ! " % + % n cap λ (1.8) %Qn(z)% → exp −V (z) 198 E. A. RAKHMANOV + in capacity in C ∼ supp λ for spherically normalized polynomials Qn = cnQn.In addition we have % % 1 % % 2 % +2 dt % cap→ −2w ∈ C ∼ (1.9) % Qn(t)fn(t) − % e z supp λ S t z where w = wϕ,S is the extremal constant for S. 1.3. The existence problem for S-curves. In a typical application of The- orem 1.3 in approximation theory function (element) f is analytic and multi-valued in Ω ∼ e where e ⊂ Ω is a small set, that is, element f has analytic continuation along any path in the domain. We denote a class of such functions by A(Ω ∼ e). We also denote by Pn set of polynomials of degree at most n. For f ∈A(Ω ∼ e)letT = Tf be a set of systems of curves Γ in Ω such that f ∈ H(Ω ∼ Γ). Let a system of polynomials Qn(z) ∈ Pn is defined by (1.7) with integration over Γ ∈T (in place of S).ImportantisthatgenerallyS ∈T with S-property is not given and its existence is not known. This leads to the problem of finding S ∈T with S-property. The context may be different from what we have used as the original motivation; the problem may not be related to complex orthogonal polynomials. In general, we have a domain Ω with a harmonic function external field in it (maybe, with a small set of singular points) and a class T of curves in Ω. We want to find a curve with S-property in T if such a curve exists. Fundamentally important is the case ϕ ≡ 0. First, S-existence problem in this case essentially includes a number of extremal problems in geometric theory of analytic functions. Second, the case is related to classical convergence problem for (diagonal) Pad´e approximants to functions with branch points. We mention some of related results in the next section. Atthesametimethecaseϕ ≡ 0 is rather specific in context of S-existence problem. Under general assumptions on class T ,anS-curve exists and is unique [56]. In the presence of a nontrivial external field, neither existence nor uniqueness are guaranteed. 2. Pad´e approximants for functions with branch points For a finite set A = {a1,...,ap}∈C of distinct points we consider an element at ∞ ∞ f (2.1) f(z)= k ∈A(C ∼ A). zk n=0 Let πn(z)=(Pn/Qn)(z) be diagonal Pad´e approximants to f, that is polynomials Pn,Qn ∈ Pn (Pn — the set of all polynomials of degree at most n) are defined by 1 (2.2) R (z):=(Q f − P )(z)=O ,z→∞ n n n zn+1 (see [5, 6] for details). First results on convergence of the sequence {πn} to f (for A ⊂ R)wereob- tained by J. Nuttall who also made the following conjecture (see [44], [46]). Let (2.3) T = {Γ ⊂ C : f ∈ H(C ∼ Γ)} ORTHOGONAL POLYNOMIALS AND S-CURVES 199 and S ∈T is defined by the minimal capacity property (2.4) cap(S)=mincap(Γ) Γ∈T Then sequence {πn} converges to f in capacity in the complement to cap πn → f, z ∈ C ∼ S The conjecture has been proven by H. Stahl [56], [57] under more general assump- tion cap(A) = 0. More exactly, he proved the following Theorem 2.1. Let cap(A)=0then + − (i) there exist and unique S ∈Tf with S-property for which w = f − f ≡ 0 on any analytic arc in S; 1 X →∗ (ii) for denominator Qn of Pad´e approximants we have n (Qn) λ where λ is equilibrium measure for S. Note that (ii) implies (1.8) and (1.9) with fn ≡ 1. H. Stahl created an original potential theoretic method for studying limit zero distribution of Pad´e denomina- tors Qn(z) based directly on orthogonality relations k Qn(z) z f(z) dz =0,k=0, 1,...,n− 1; Γ ∈Tf . Γ for these polynomials written with Γ = S where S has property (1.6). The method was further developed in [24] for the case of presence nontrivial external field; Theorem 1.3 has been proved using this development. It is important to observe that according to assertion (i) of the theorem an compact S-compact always exists (there is no “if” in the theorem). For a finite set A this part of the theorem is close to well-known Chebotarev’s problem in geometric function theory. Chebotarev’s problem was the problems of existence and characterization of a continuum of minimal capacity containing A. It was solved independently by Gr¨otzsch and Lavrentiev in the 1930s. The following theorem is just one example of a large class of theorems presenting solutions of extremal problems in function theory in terms of quadratic differentials (for details we refer to [32]) and [60]. Theorem 2.2. For a given set A = {a1,...,ap} of p ≥ 2 distinct points in C there exist a unique set S cap(S)=mincap(Γ) Γ∈T where T is the class of continua Γ ⊂ C with A ⊂ Γ. The complex Green function G(z)=G(z,∞) for C ∼ S is given by z p−2 G(z)= V (t)/A(t) dt, V (z)=z + ···∈Pp−2. a where A(z)=(z − a1) ...(z − ap) and V is uniquely defined by A. Thus, we have S = {z :ReG(z)=0} so that S is a union of some of critical trajectories of quadratic differential (V/A)(dz)2. It is also the zero level of green function g(z)=ReG(z) of two-sheeted Riemann surface for V/A. 200 E. A. RAKHMANOV Stahl proved that assertions of Theorem 2.2 remain valid for S in Theorem 2.1. In his recent paper [58] he introduced a more general concept of “extremal cuts” (or maximal domain) related to an element f of an analytic function at ∞.Let f ∈A C ∼ e where e is a compact of positive capacity. Let Tf = T is defined by (2.3) above. The main result in [58] is the existence and uniqueness theorem of acompactS ⊂Tf with (2.4).In general such S does not have an S-property (but contains a part with this property). The paper contains also an extended review related to the standard case cap(e)=0. 3. An existence theorem for an S-curve in harmonic external field. We formulate conditions of existence in terms of Hausdorff metric and local variations; terms are explained in Sections 3.1 and 3.2 next. 3.1. Hausdorff metric. Classes of compacts. Let −1/2 −1/2 2 2 d (z1,z2)=|z1 − z2| 1+|z1| 1+|z2| be the chordal distance in C. For two compacts K1,K2 ⊂ C their Hausdorff distance δH is defined as { ⊂ ⊂ } (3.1) δH (K1K2) = inf δ>0:K1 (K2)δ ,K2 (K1)δ where (K) − δ–neighborhood of K in the chordal metric: δ (3.2) (K)δ = z ∈ C :mind(z,ζ) <δ . ζ∈K In what follows we consider mainly compacts which are finite unions of continua. For such compacts K we denote by s(K) the number of connected components of K. Let Ω be a domain in C, then set of all compacts K ⊂ Ωisacompactin δH -metric. The same is true for all compacts K ⊂ Ω with s(K) ≤ s. We use chordal metric on C and associated Hausdorff metric δH only to formu- late Theorem 3.1 below. In the proof of the theorem in Section 9 we immediately reduce consideration of compacts K ⊂ C to the case K ⊂ C; so that we can use Euclidean distance d (z1,z2)=|z1 − z2|, associated neighbourhoods (K)δ and cor- responding Hausdorff distance δH . Metrics are equivalent on compacts in any fixed disc in the open plane. 3.2. Local variations. For a closed set A ⊂ C and a complex valued functions h(z) ∈ C1 C with h(z)=0,z∈A. we introduce one parametric family of local variations with fixed set A in the direction of h, that is, transformations of C defined by (3.3) z → zt = z + th(z),t≥ 0 We also call them A-variations (in the direction of h). For small enough t transfor- mations (3.3) are one-to-one; in a standard way they generate variations of compacts Kt and measures μt (3.4) K → Kt = {zt : z ∈ K},dμt zt = dμ(z). Variations defined above leave the point ∞ unchanged; it is always technically convenient to include infinity into the fixed set. In many cases we are interested ORTHOGONAL POLYNOMIALS AND S-CURVES 201 in variations of a particular compact K, then the condition h(z) ∈ C1 C may be replaced by h(z) ∈ C1 (Ω) where Ω is a domain containing K. It is important to note that in many cases we use more restricted classes of A-variations; in particular, we often use functions h satisfying a stronger condition h = 0 in a neighbourhood of A (see Remark 4.1). 3.3. Main Theorem. The following is a version of an existence theorem. Theorem 3.1. Suppose that the external field ϕ and class of curves T satisfy the following conditions: (i) ϕ is harmonic in C ∼ e where e ⊂ C is a finite set (ii) (a) T is closed in a δH -metric (b) For a finite set A ⊂ C, T is open in topology of A-variations: Γ ∈T implies Γt ∈T for small enough t for any A-variation. (c) For some natural s any Γ ∈T has at most s connected components; (iii) (a) There exist Γ ∈T with Eϕ[Γ] = inf Eϕ(μ) > −∞; μ∈M(Γ) (b) Eϕ{T } =supEϕ[Γ] < +∞. Γ∈T (iv) For any sequence Γn ∈T that converges in δH metric (δH (Γn, Γ) → 0) there exists a disc D such that Γn ∼ D ∈T Then there exists a compact S ⊂T with the S-property. Moreover, for the equilibrium measure λ = λϕ,S of the compact S the following condition is satisfied dλ(t) 2 (3.5) R(z)= +Φ(z) ∈ H(Ω ∼ A ∪ e) t − z that is, the function R(z) is holomorphic (analytic and single valued) in Ω ∼ (e ∪ A)=Ω (ϕ =ReΦ). Furthermore, R has simple poles at each point a ∈ A ∼ e (singularities at points in e are essentially those of (Φ)2). Finally, supp λ us a union of analytic arcs which are trajectories of quadratic differential R(z)(dz)2√and this differential is the one with closed trajectories; we 1 | | also have dλ(z)= π Rdz along arcs in supp λ. A short outline of the proof is presented in Section 3.4; details are contained in Section 9. Particular versions of the theorem and parts of related techniques were discussed in [24, 49, 30, 37]. Here we make a few general remarks. Points of sets e and A are both singularities of R(z) in (3.5) above; but points from A ∼ e are simple poles of R, singularities of R(z)atpointsine are same as in Φ(z)2. Besides that there is not much difference between sets e and A —both will eventually be fixed points of variations and will often unify them in single set A. Theorem 3.1 remains valid if both sets A and e are closed sets of capacity zero. Case of finite sets is, probably, the most interesting, since R(z) in this case has a finite number of singular points in C and in many cases may be explicitly found. On the other hand sets of singularities of positive capacity present an essential problem which we do not discuss in this paper. 202 E. A. RAKHMANOV In the connection with basic properties (ii) of class of curves T we note that associated assumptions may be stated in different terms. We have selected an abstract way which has probably some advantage in generality. The following example is fundamental. Let f ∈A C ∼ A and let Tf be the class (2.3) of admissible cuts Γ associated with f. In many applications the class T may be directly or indirectly defined as Tf with some f ∈A C ∼ A . It is important to observe that S-curve exists in Tf but the class does not formally satisfy conditions of Theorem 3.1. It is not closed in δH -metric since the definition contain an implicit condition that the point where element f is defined (in the current case it is ∞) is not in any Γ. The situation is typical; many important classes of curves are not closed. However, in most of naturally arising situations the maximizing sequence converges to an element which belong to Tf .Moreover, it is usually not difficult to prove it; we will have a few examples below. Alternatively we may define T as collection of finite unions Γ = ∪Γj of con- tinua Γj ; each of them connects certain groups of points in A or separates one group from others. Such classes T would satisfy conditions of the theorem if we do not have exceptional points). In this connection we note that some particu- lar S-problems (especially with rational external field) are similar to “extremal partitions” or “moduli” problems in geometric function theory; see [32], [60]and also [37]. Theorem 3.1 will not be valid without condition (ii)(c) since any compact may be approximated in δH -metric by a finite number of points. Continuity of equilib- rium energy is not preserved in such circumstances. We present a few examples related to conditions (iii). Let T be class of continua connecting points 0 and 1 and ϕ =logz then S-compact does not exist since Eϕ[Γ] = −∞ for any Γ ∈T (in connection with possible applications it is more reasonable to include curves Γ bypassing zero; then (iii)(a) is satisfied and S-curve exists). In another typical example T is set of Jordan contours separating zero and infinity ϕ = a log z, a>0. If a =1 /2wehaveEϕ{T } = ∞ so that the condition (iii)(b) is not satisfied and S-curve does not exist (if a<1/2 a maximizing sequence Γn collapses to zero; if a>1/2 it collapses to ∞). If a =1/2 then any circle centered ar zero is an S-curve which is an example of non-uniqueness. Condition (iv) is technical. Haussdorff closure always contains large sets which are of no interest in the connection with maximization for equilibrium energy. For instance, let T be the family of analytic arcs containing two fixed points. Then its closure T in Haussdorff metric contains Γ = C. Without condition (iv) such possibilities will require unnecessary separate consideration. 3.4. Outline of the proof: max-min energy method. The method has two components. First, we study continuity properties of the equilibrium energy functional (3.6) Eϕ[Γ] = inf Eϕ(μ):T→[−∞, +∞] μ∈M(Γ) on T with the Hausdorff metric.The main result here is the following Theorem 3.2. Functional Eϕ above is upper semi-continuous. We present the proof of the theorem in Section 9 where we introduce a larger class K of compacts K in C with a bounded number of connected components. ORTHOGONAL POLYNOMIALS AND S-CURVES 203 First we consider continuous external field ϕ (not supposed to be harmonic) and prove that the functional Eϕ : K→(−∞, +∞) is continuous. Then we use continuous fields ϕ to approximate fields with singular points. Theorem 3.2 is actually the most extended part of the proof; the rest of it is comparatively easy reduced to known results. An immediate corollary of the theorem is the following Corollary 3.3. There exist the extremal compact S with (3.7) Eϕ[S]=supEϕ[Γ]; S ∈T. Γ⊂T The second component of the method is a method of energy variations devel- oped in [49], [30]and[37] which lead to the following Theorem 3.4. The compact S in (3.7) above has S-property. The only part of Theorem 3.4 which requires a proof is the following. Lemma 3.5. Equilibrium measure of extremal compact S in (3.7) is a critical measure associated with external field ϕ and fixed set A. The fact that total potential of a critical measure satisfies S-property is known. We go into some details in the next section. Thus, Lemma 3.5 concludes the proof of existence theorem. 4. Critical measures and equilibrium measures of S-curves 4.1. Main definitions. Let Ω ⊂ C be a domain, A ⊂ Ω be a finite set and ϕ be a harmonic function in Ω ∼ A. For an A-variation z → zt = z + th(z)inΩ∼ A we define the associated variation of weighted energy of a measure μ in Ω by 1 t (4.1) DhE(μ) = lim Eϕ μ −Eϕ(μ) . t→0+ t (μt is defined in (3.4).) We say that μ is (A, ϕ)-critical if for any A-variation such that the limit above exists we have (4.2) DhE(μ)=0. There are two important facts about critical measures. First, the equilibrium measure of an S-curve is a critical measure. Second, the potential of a critical mea- sure λ has S-property (1.6). Besides, critical measures are often more convenient to deal with then S-curves. Critical (stationary) measures were first introduced in [24] and then used in [49] and later in [30] in combination with min-max method. A systematic study of critical measures for rational field was carried out in [37]. The last paper contains all the facts related to critical measures which we need here (a few cosmetic changes required). We single out some important formulas leading to S-property of a critical measure. First, we have the following explicit representation [37]. h(x) − h(y) (4.3) D E(μ)=Re dμ(x)dμ(y) − 2 Φ(x)h(x)dμ(x) h x − y where ϕ(z)=ReΦ(z); we have Φ ∈ H(Ω ∼ A). 204 E. A. RAKHMANOV Next, lim in (4.1) exists iff Φ h ∈ L1(μ). For technical reasons it make sense to replace the last condition with a more restrictive one h(z)=0forz ∈ (A)δ with some δ>0 and it is convenient to include it in the definition of critical measure (see Remark 4.1). Next, for any (A, ϕ) critical measure μ we have dμ(t) 2 (4.4) R(z)= +Φ(z) ∈ H(Ω ∼ A) t − z (Lemmas 2 and 3 in [37]). In turn, this assertion implies the following descrip- tion of Γ = supp μ: Γ consists of a finite number of critical or closed trajectories of quadratic differential −R(dz)2. Moreover, the last differential is closed (more exactly, it is quadratic differential with closed trajectories [60]), [37] Theorem 5.1. Together with (4.4) this yields representation z (4.5) V μ(z)+ϕ(z)=−Re R(t) dt, z ∈ C ∼ Γ. a √ 1 | | The S-property of Γ together with the formula dμ(z)= π Rdz on open arc of Γ follow directly from (4.5). Remark 4.1. Actually, to derive properties (4.4) and (4.5) of a critical measure one needs to verify (4.2) for some particular class of functions h . More exactly, it was shown in [37] that it is enough to verify (4.2) for modified Schiffer’s variations h(z)=θ(z)A(z)/(z − ζ)whereA is the polynomial whose roots are fixed points, ζ is a complex parameter and θ(z) is a real function which is equal to zero in a neighbourhood of points from A and it is equals to unity outside of a slightly larger neighborhoods. We do not have to go into such details; instead we define critical measures as a measures μ with (4.2) valid for any smooth h(z) satisfying the condition h(z)=0forz ∈ (A)δ with some δ>0. This class is larger but easier to describe. We note that class of (unit positive) critical measure is larger than class of equilibrium measure of S-compact (for the same external field and same fixed set). Indeed, for the equilibrium measure λ = λϕ,S we have relations (1.5) and (1.6); in particular, V λ + ϕ is a constant on the supp λ (besides, there is a part S ∼ supp λ of S which has in general certain degree of freedom). Thus, we have two closely related notions (1) S-curve in a (homotopic) class T with fixed set A in the external field ϕ; (2) (A, ϕ)-critical measure μ. Next we make some more comments in this direction for the (classical) case ϕ ≡ 0; see [37] for details. 4.2. A-critical measures and S-curves for ϕ ≡ 0. We compare critical measures and equilibrium measures of S-curves without external field and with the same fixed set A = {a1,...,ap}. Let classes of curves be introduced as Tf associated with various functions f ∈A(C ∼ A). For any (unit positive) A-critical measure p,−2 ,p there exists a polynomial V (z)= (z − vj ) such that with A(z)= (z − ak) j=1 k=1 (here we denote polynomial and its set of zeros by same symbol) we have % % z 1 % % (4.6) V μ(z)=Re V (t)/A(t) dt, dμ(z)= % V/Adz% . a1 π ORTHOGONAL POLYNOMIALS AND S-CURVES 205 (in particular, supp μ is a union of critical trajectories of (V (z)/A(z)) (dz)2). The same is true for the equilibrium measures of S-curves in class Tf . Thus, both sets of measures may be characterized in terms of associated polynomials V .Now,using zeros vj of V as parameters, we describe set of A-critical measures ( [37], Section 9). Space of vectors VA = {v =(v1,...,vp−2)} corresponding to critical measures has real dimension p − 2; more exactly, it is a union of 3p−2 bordered (bounded) domains (cells) on a manifold of real dimension p − 2. Interior points of each cell correspond to measures μ with Γ = supp μ consisting of exactly p−1 simple disjoint p−2 analytic arcs Γj with endpoints from {ak,vj }. As parts of C cells (corresponding v ∈ Cp−2) are defined by systems of equations (4.7) Re V (t)/A(t) dt =0,j=1,...,p− 2. Γj On the other hand there is only a finite number of S-curves associated with a givenfixedsetA and any of them may be obtained as S-curves associated with a class Tf of admissible cut for a properly selected functions f ∈A(C ∼ A). Equilib- rium measures of S-curves are among A-critical measures and they are located on boundaries of cells. In particular, v0 corresponding to the Chebotarev’s continuum T T 0 = A for A belongs to the boundary of each cell (and may be used for a more explicit description of VA). Further, equilibrium measures of S-curves satisfy (4.7) and also p−2 additional equations which distinguish them among critical measures. These additional equations may be presented as follows. For any critical measure μ with disjoint arcs Γj of support of μ we have ⎛ ⎞ p3−1 μ ⎝ ⎠ V (z)=Cj,z∈ Γj supp μ = Γj . j=1 Additional equations for an equilibrium measure of an S compact are C1 = ···= Cp−1. These equations may also be written in the form (4.7) so, all of them are equations on real parts of periods of quadratic differential V/A dz2 . We have totally 2p − 2 real equations for the same number of real parameters in V .Thissystemof equations defining equilibrium measures of S-curves has rather complicated analytic structure. Critical measures may be useful in this context since they constitute a connected space. 4.3. Heine–Stieltjes and generalized Jacobi polynomials. Critical mea- sures were studied in [37] with the purpose of characterization of zero distributions ,p of Heine–Stieltjes polynomial. For a fixed A(z)= (z − ak) with distinct ak k=1 − p−1 n + p 1 and B(z)=αz + ··· ∈ P − and a fixed n ∈ N there exist δ = p 1 n n polynomials Vn(z) ∈ Pp−2 (Van Vleck polynomials) such that differential equation (4.8) A(z)y (z)+B(z)y (z) − n(n + α − 1)Vn(z)y(z)=0 n has a polynomial solution y(z)=Qn(z)=z ... of degree n (Heine–Stieltjes poly- nomials). One of the main results in [37] is the following. If we have a convergent 206 E. A. RAKHMANOV p−2 sequence of Van Vleck polynomials Vn(z) → V (z)=z +··· then for correspond- ing Heine–Stieltjes polynomials we have 1 ∗ X (Q ) → μ n n where μ is an A-critical measure. Moreover any A-critical measure may be obtained 1 X this way. In electrostatic terms n (Qn) is a discrete critical measure and the result simply means that a weak limit of a sequence of discrete critical measures is a (continuous) critical measure. It is interesting to compare Heine-Stieltjes polynomials with Generalized Ja- cobi polynomials — denominators of diagonal Pad´e approximants at infinity for functions p 1 z B(t) f(z)=exp dt = (z − a )αk 2 A(t) k k=1 (assume that α1 + ··· + αk =0;thenf is analytic at ∞). Let Pn(z)bethe corresponding Pad´e denominator. It is a classical fact that for each n ∈ N there p −2 exists a polynomial hn(z)=z + ···∈ Pp −2 where p ≤ p such that y = Pn(z) is a solution of the Laguerre equation − − (4.9) Ahny +(A hn Ahn + Bhn) y n(n +1)Cny =0 2p −4 where Cn(z)=z + ··· ∈ P2p −4. Comparison of (4.8) and (4.9) shows that Pn(z) are also Heine–Stieltjes polynomials with redefined A and B. 1 X →∗ Note that by Stahl’s theorem we have n (Pn) λ where λ is the Robin measure for S-compact S ∈Tf . Thus, Heine–Stieltjes and generalized Jacobi poly- nomials present discrete versions of critical measures and equilibrium measures of S-curves respectively. Strong asymptotics for GJ polynomials Pn based on the equation (4.9) were obtained by J. Nuttall [47]forp = 3; recently the result was extended for arbitrary p [38]. Strong asymptotics for HS polynomials Qn based on the equation (4.9) were obtained in [36]. Related formulas may be interpreted as solution of an S-problem; we will go into some details in Section 8 below. 5. Rational external fields In Section 4.2 above we discussed equilibrium measures of S-curves and critical measures associated with a fixed set A = {a1,...,ap} in a classical non-weighted case. Now we introduce an important class of rational external fields and single out a few corollaries of Theorem 3.1. 5.1. Field of a system of fixed charges in plane. We place a (real) charge αk at the fixed point ak. This system of charges generate the external field ϕ(z)= p αk αk log(1/|z − ak|). Then corresponding Φ = − is rational function (so z−ak k=1 that we consider the case as rational). Let Tf be class of curves associated with a function f ∈A C ∼ A (recall that in such situations we admit that Γ ∈T may bypass ak with negative αk). Theorem 3.1 implies that there is a compact Sf ∈Tf with S-property. Subsequently it follows by (4.4) that corresponding R is a rational function with second order poles at points ORTHOGONAL POLYNOMIALS AND S-CURVES 207 2p,−2 from A and seconds order zero at ∞. So, there is a polynomial V (z)= (z − vk) k=1 such that for the total potential of equilibrium measure λ we have % % % % z V (t) 1 % V (z) % λ % % (5.1) V + ϕ (z)=Re dt; dλ(z)= % dz% a1 A(t) π A(z) The same is true for any (A, ϕ) critical measure μ. Thus, both families of all (unit, positive) critical measures and equilibrium distributions of S-compacts in the field 2p−2 ϕ are described as subsets of space C = {v =(v1,...,v2p−2)} just as for the case ϕ ≡ 0 in Section 4 above. Furthermore, similar to what we have in case ϕ ≡ 0, the set of (A, ϕ)-critical p−2 − measure is a union of cells — bordered manifolds Mj of real dimension p 2 which may be locally represented using v ∈ C2p−2 as local parameters by a system of equations including p − 2realequations (5.2) Re V (t)/A(t) dt =0,j=1,...,p− 2 Γj and p complex equation (5.3) V (aj )/A (aj )=αj ,j=1,...,p √ (Equation (5.3) prescribes residues of V/Aat poles). We formally state what we said above as Theorem 5.1 below (joint result with A. Mart´ınez-Finkelshtein). p Theorem 5.1. Let ϕ(z)= αk log(1/|z − ak|), αk ∈ R. Then set of (A, ϕ)- k=1 critical measures is a union of finite number of cells — bordered manifolds of real dimension p−2. Interior points of each cell correspond to measures μ whose support is a union of p − 1 disjoint analytic arcs Γj . Endpoints of those arcs belong to {a1,...,ap,v1,...,v2p−2) and satisfy equations (5.2) and (5.3). Extra p − 2 equations which distinguish equilibrium measures of S-curves in various classes Tf are similar to those in Section 4. The key point in the proof of the theorem is existence of weighted Chebotarev’s continuum associated with the ϕ which is the compact S with S-property is class T 0 of continua containing A in the external field ϕ (see Theorem 3.1). As a corollary of the theorem 5.1 one can obtain extentions of the theorem on zero distribution of Heine-Stieltjes polynomials (4.8) from [37] discussed in 4.3 to a larger class with B = Bn depending on n in such a way that there exist a limit 1 lim Bn(z). Such polynomials are interesting in the connection with so called n→∞ n Gaudin’s model (see [42]; see also [39] for an earlier “real” version of the theorem). Subsequently, methods of [36] may be used to obtain strong asymptotics for those polynomials. 5.2. Multi-point Pad´e approximants. As another application of Theorem 3.1, we consider a problem of interpolation of several (generally different) analytic elements by a single rational function with free poles. Interpolation points are defined by the set A = {a1,...,ap}⊂C,thenforeachk =1,...,p we introduce a finite set sets of distinct points Ak ⊂ C (branch points of functions); sets Ak may 208 E. A. RAKHMANOV intersect. Next, let fk be a function elements at ak with branch points from Ak ∞ n fk(z)= ck,n (z − ak) ∈A C ∼ Ak ,k=1,...,p. n=0 Finally, let mk,n ∈ Z+, k =1,...,p, n ∈ N be nonnegative integers satisfying conditions p mk,n mk,n =2n +1, lim = mk,k=1,...,p n→∞ n k=1 (so that m1 + ···+ mp = 2). Then we can find a rational function πn = Pn/Qn of order ≤ n interpolating fk at ak with multiplicity mk,n. Polynomials Pn,Qn are defined by conditions mk,n (Qnfk − Pn)(z)=O ((z − ak) )asz → ak (such polynomials Pn,Qn ∈ Pn exist and the ratio Pn/Qn is unique). The problem of convergence and the sequence {πn} reduces to the problem of zero distribution for polynomials Qn. Those polynomials may be defined by or- thogonality conditions with weights including varying parts 1/Ωn(z)whereΩn(z)= ,n mk,n (z − ak) . k=1 Investigation of the zero distribution of Qn requires (but is not completely reduced to) the existence of an S-curve in the class T of curves Γ0 ∪ Γ1 ∪···∪Γp ⊂ C ∼ A with the following properties: C ∼ Γ0 =Ω1 ∪ Ω2 ∪···∪Ωp; ak ∈ Ωk;Ωi ∩ Ωj = ∅; Γk ⊂ Ωk,fk ∈ H (Ωk ∼ Γk) and the additional condition that the jump of f over any analytic arc in Ω is ≡ 0. Now, there are certain exceptional situations when conditions of Theorem 3.1 are not satisfied and the required S-curve does not exist. For instance it happen when we interpolate two different constants at two different points (see example in Section 3.3). We do not go into further details; normally conditions of the Theorem 3.1 are satisfied, S-curve exists and may also be defined by the extremal property p Eϕ[S]=maxEϕ[Γ] with the external field ϕ(z)= mk log |z − zk|. ∈T Γ k=1 However, in many cases support of the equilibrium measure related to the problem is disconnected and Theorem 1.3 is not directly applied. A way around was found in [14] for a particular case of two-point approximation for two elements with two branch points each. We note also that if the poles of the interpolating function Fn (zeros of Qn) are partially fixed then we come to a problem more general then the one above; external field has an additional positive charge which makes situation a little more complicated. 5.3. Polynomial external field. Let p(z)=czm + ··· be a polynomial of degree m ≥ 2. Let T be a class of curves Γ = Γ1 ∪ Γ2 ∪···∪Γp where Γj is a Jordan arc in C which goes from ∞ to ∞. More exactly, each Γj begins and ends in different sectors where ϕ(z)=Rep(z) →∞in such a way that the integrals in k −2np(z) Qn(z)z e dz =0,k=0, 1,...,n− 1 Γ ORTHOGONAL POLYNOMIALS AND S-CURVES 209 exist (and not trivially vanish for any polynomials Qn.) Then the relations above n define a sequence of polynomials Qn(z)=z +···. Then, by Theorem 1.3, we have 1 → ∈T n χ (Qn) λ,whereλ is the equilibrium measure of the S-curve in ϕ with S if such a curve exists. Conditions of Theorem 3.1 are not satisfied; indeed, set of singularities (which is also a fixed set) consists of a single point ∞ but class T is not closed (for instance, {∞} ∈ T Γ= S is a limit point of in the spherical δH metric). However, it is (n) (n) not difficult to show that for any Γ → Γ ∈T we have Eϕ Γ →−∞and, therefore, maximizing Γ(n) has a limit point in Γ. By Theorem 3.1, this limit point has the S-property. An interesting problem is to characterize polynomials R(z) which may be ob- tained this way and, thus, characterize corresponding S-curves. Some progress in this direction was made in [10]. One parametric family p(z)=tz2 +z4 is considered in [12]. The work [35] is in progress where equilibrium measures on R in the field of ϕ =Rep, p ∈ P4 are considered. The problem is now far from being completely solved. We return to the existence problem for external field ϕ =Rep: There is an- other way to arrange the reference to Theorem 3.1 which may have an independent interest. In short, we can cut off curves Γ = ∪Γj ∈T taking intersection with a disc of large radius and subsequently take the limit as radius tends to infinity. More exactly, each Γj is defined by two different sectors of plane. Let θ1,θ2 be arguments iθ1 iθ2 of middle lines of those sectors. Define aj,n = ne , bj,n = ne :andletΓj,n be any curve in plane connecting aj,n and bj,n.LetΓn be class of curves Γn = ∪Γj,n; this class satisfies condition of Theorem 3.1 and there exists an S-compact Sn ∈Tn. For its equilibrium measure λn we have dλ (t) 2 q (z) R (z)= n + p(z) = n n t − z s (z) n where sn(z)= (z − aj,n)(z − bj,n)andq(z) ∈ Pm where m =2m − 2+2p.We j note that S-curves and measure λn associated with the situation have independent interest if endpoints are prescribed in advance. In the original context, as those points tends to ∞ the function Rn will become a polynomial as n ≥ N (qn is divisible by Sn) and corresponding Sn maybe completed arcs going to ∞ to a curve S of original problem. The situation discussed above is typical: in many applications conditions of Theorem 3.1 are not immediately satisfied, but reduction is eventually possible. 6. Green’s and the vector equilibrium problems Each equilibrium problem for logarithmic potential has an analogue for the Green’s potential. Most part of corresponding definitions and assertions are essen- tially identical. We make a brief review of them next. 6.1. Definitions. Let Ω be a domain and μ be a positive Borel measure in Ω. We denote by μ ∈ (6.1) VΩ (z)= g(z,t) dμ(t),zΩ the Green’s potential of μ (g(z,ζ) is the Green function for Ω with a pole at z = ζ). This class of potentials is in many ways simpler than class of logarithmic potential. 210 E. A. RAKHMANOV μ The function V = VΩ is invariant under conformal mappings of Ω (it may be defined as solution of Poisson equation ΔV = −2πμ with boundary values V =0on∂Ω). We have for any positive μ in Ω μ ≥ E Ω μ ≥ VΩ (z) 0; (μ)= VΩ dμ 0. For a compact F ⊂ Ω and an external field ϕ ∈ C(F ) there exist a unique equilib- rium measure λ on F defined by each of the conditions (1.4) and (1.5) where Eϕ λ E Ω λ and V have to be replaced by ϕ and VΩ . Next, let a ϕ be a harmonic function in Ω ∼ e (cap(e) = 0). We say that F has λF λ Green’s S-property (relative to Ω) and ϕ) if (1.6) is valid for VΩ in place of V . Problem of existence of a compact with (Green’s) S-property we will abbreviate to (Green’s) S-problem. As for logarithmic potential, the case ϕ = 0 is classical. It is customary in this case to set E = C ∼ Ω and consider a pair of disjoint compacts (condenser) (E,F) instead of compact F in the domain Ω = C ∼ E. Capacity C(E,F) of a condenser (E,F) is defined by 1 1 C (E,F)= Ω = E (λF ) E (λF − λE) where Ω = C ∼ E, λF -Green’s equilibrium measure for F (with ϕ ≡ 0); λE is balayage of λF onto ∂Ω ⊂ E.Signedmeasureλ = λF − λE is called equilibrium measure of condenser (E,F). We note that E and F are interchangeable in this context, in particular C(E,F)=C(F, E). In the presence of an external field we assume that it is acting on F . We note also that λ = λF − λE may also be interpreted as a vector measure λ =(λE,λF ); we will go into some detail later in Section 6.4. The equilibrium problems for Green’s potential (S-problem, in particular) play, first of all, an important role in the theory of best rational approximations to analytic functions. Next we review briefly classical results by J. Walsh, A. Gonchar and H. Stahl related to the case ϕ =0. 2 6.2. Gonchar’s ρ -conjecture. Let E ⊂ C be continuum and f = fE be an element of analytic function on E.LetRn be the set of all rational functions rn = Pn/Qn of order ≤ n (Pn,Qn ∈ Pn)and ρn(f)= minmax |f(z) − r(z)|. r∈Rn z∈E A well-known theorem of the 1930s proposed by J. Walsh asserts that 1 −1/C(E,F) (6.2) lim ρn(f) n ≤ ρ(f) = inf e . n→∞ F ∈F Where F is the set of all compacts F ⊂ C ∼ E such that f has analytic continuation from E to Ω = C ∼ F (see [64] for further details). In 1978, A. Gonchar [20](seealso[19]) studied approximations for Markov- type functions f on an interval E of the real axis and observed that it is possible in this case to replace ρ(f) in Walsh’s estimate with ρ(f)2. Later, he generalized this result and proved also that in many important cases lim may be replaced with lim and ≤ may be changed to = (see [20] for details and further references). Thus, ORTHOGONAL POLYNOMIALS AND S-CURVES 211 for some important classes of functions equality holds 1 2 (6.3) lim ρn(f) n = ρ(f) n→∞ So, Gonchar made his so-called called ρ2-conjecture: let for some compact e ⊂ C ∼ E,cap(e) = 0 function f|E has analytic continuation along any path in C ∼ e. Then (6.3) is satisfied. The upper bound in this conjecture (with lim) was proved in 1986 by Stahl [57]. The associated low bound and, therefore, existence of lim was established a year later in [24]. So, the ρ2-conjecture becomes a theorem. Proof of the Gonchar–Stahl theorem was based on a general version of multi- point Pad´e approximants (free poles interpolation). In short it may be presented as follows: Let Ff be the class of compact F ⊂ Ω with f ∈ H C ∼ F .There exists a compact S ⊂Ff with Green’s S-property relative to Ω [57]; let λF be its Green’s equilibrium measure. Finally, let λE be the balayage of λF onto ∂Ω ⊂ E. 2n Let Gn(z)=z + ··· ∈ P2n be a sequence of polynomials with zeros on E 1 X →∗ →∞ ∈ R such that 2n (Gn) λE as h .Letπn(z)=Pn/Qn n be corresponding sequence of (linear) Pad´e approximants to f that is (Qnf − Pn)/Gn is analytic on E. Then denominators Qn of approximations πn satisfy orthogonality relations k f(z) (6.4) Qn(z)z dz =0,k=0, 1,...,n− 1. F Gn(z) 1 X →∗ which is (1.7) with fn = f/Gn and by Theorem 1.3 it follows that n (Qn) λF . Then (1.9) together with Hermite interpolation formula imply (2.3) (Theorem 1.3 for this case was actually proved in [57]). The key point in the construction of the proof above is investigation of complex orthogonal polynomials Qn which was possible to carry out using Theorem 1.3 since associated Green’s S-problem without external field has always a positive solution [56] (we note that this Green’s problem is equivalent to some logarithmic S-problem with external field generated by a negative charge. A more general situation is related to interpolation of an analytic element f with small set e of branch points on a set E according to an arbitrary interpolation table on E.CasewhenE and e are separated by a circle is considered in [7] where, in particular, associated S-problem is solved. An inverse problem of finding harmonic field in which a given analytic arc has S-property is considered in [8]. 6.3. Best rational approximations to the exponential on R+. More general theorem on the best rational approximation to a sequence of analytic func- tions was proved in [24]. The proof followed the same path as in the proof of Gonchar–Stahl theorem above which leads to the complex orthogonality relations for denominators of related Pad´e Approximants. The formula of n-th root asymp- totics is then needed for these polynomials which requires solution of an S-problem with a harmonic external field. It was the context in which S-problem has been for the first time explicitly introduced; hypotheses of the general theorem included an assumption that related S-curve for Green’s potential exists in a given class. 1 Then, for an important particular case related to a solution of the well-known “ 9 - problem” the associated S-problem was constructively solved and a few important remarks were made about the problem in general. 212 E. A. RAKHMANOV −x Let ρn =minmax |e − r(x)|. The problem of associated rate of convergence r∈Rn x∈R+ was later called the “1/9-problem” by E. Saff and R. Varga [53], since numerical 1/n experiments showed that ρn ≈ 1/9 for large enough n. Originally, the problem was introduced in the end of 1960’s in connection with the numerical solution of the heat-conducting equation and it attracted common attention. It was eventually solved in [24], where one can also find a review of the preceding results. The method described above lead the following equilibrium problem. Let Ω = C ∼ R+, F is a class of curves F in Ω which go from ∞ to ∞ around R+ in such awaythatϕ(z)=Rez → +∞ as z →∞, z ∈ F (actually we need ϕ → +∞ fast enough, a fact which presents some technical problems); instead we may consider curves F from a − i to a + i in Ω where a>0 is large enough. Now the question was if there is a Green’s S-curve S ∈T in the field ϕ relative to Ω. The positive answer has been obtained in [24] by construction of S-curve and, subsequently the rate of convergence of the best approximants has been found in terms of the parameters of its equilibrium distribution. More exactly, let λ = λS − λE be the λ equilibrium charge of (S, E)(hereE = R+)andw = V + ϕ (ξ), ξ ∈ supp λ be the corresponding Green’s equilibrium constant. Then 1/n −2w 1 lim ρn = ρ = e = . n→∞ 9.2 ... Furthermore, near-best approximants may obtain by interpolation of e−2nz with density represented by λE; λS represents (contracted) zero distribution of denomi- nators Qn. Strong asymptotics for polynomials Qn has been later obtained by A. Aptekarev [2] who used steepest descent method for the Matrix Riemann–Hilbert (MRH) problem; as a corollary he proved A. Magnus’ conjecture on existence and value n of lim ρn/ρ . Steepest descent as a general method of solving certain class of n→∞ MRH problems has been developed by P. Deift and X. Zhou [17](seealsothe book [15]). The method is widely used in various classes of problems; in particular, it is a powerful method for proving formulas of strong asymptotics for complex orthogonal polynomials. In a general context the method requires existence of an S-curve as well as the potential theoretic method we were discussing above. Some progress in the investigation of Green’s S-problem has been made in the paper by S. Kamvissis and author [30] where the max–min energy method has been outlined in a particular situation related to a NLS (we will go into some details in Section 6.6 below). In the next section a general existence theorem for Green’s S- problem is presented similar to Theorem 3.1 above. Actually we can carry Theorem 3.1 over to the Green’s case without essential modifications. Some of the required changes in assertions and proofs are briefly discussed next. 6.4. An existence theorem for Green’s S-curves. We preserve the basic hypotheses of Theorem 3.1. That is, we assume that the external field has finite number of singular point, the class of compacts has a finite number of fixed points (now we combine the two sets and use a single set A) and number of components in each compact is bounded by a common constant. A new detail in Green’s case is the presence of the boundary ∂Ω of domain Ω whose Green function defines potentials. We will assume that ∂Ω has finite number of connected components. Since the whole problem is conformal invariant it may be reduced to the case of a circular domain (∂Ω is a union of disjoint circles.) ORTHOGONAL POLYNOMIALS AND S-CURVES 213 Next, we assume that there is a larger domain Ω0 ⊃ Ω such that ϕ is harmonic in Ω0 except for a finite set of points (the condition may, of course, be relaxed, but we do not go here into further discussion). Finally, conditions (ii)(a) and (ii)(b) cannot be both satisfied for a family of compacts in a domain with large boundary. So, we will replace (ii)(b) with a condition on one-sided variations. This will make it possible to prove the next theorem by essentially the same way we prove Theorem 3.1 in Section 9 below. Theorem 6.1. Let Ω be a circular domain, Ω0 ⊃ Ω. Suppose that the external field ϕ and class of curves F ⊂ Ω satisfy the following conditions: (i) ϕ is harmonic in Ω0 ∼ A where A ⊂ Ω0 is a finite set (ii) (a) F is closed in a δH -metric (b) F ∈Fimplies F t ∈Ffor small enough t for any A-variation if F t ⊂ Ω (c) For some s>0 we have s(F ) ≤ s for any F ∈F; (iii) (a) There exist F ∈F with E Ω −∞ ϕ [F ] = inf Eϕ(μ) > ; μ∈M(F ) E Ω{ } ∞ (b) ϕ F =supEϕ[F ] < + . F ∈F (iv) For any sequence Fn ∈F that converges in δH metric (δH (Fn,F) → 0) there exists a disc D such that Fn ∼ D ∈F Then there exists a compact S ⊂F with the S-property. An analogue of representation (3.5) is valid for the potential of Green’s equi- librium measure λ of S (external field has to be modified; in case when Ω is upper half-plane one has to add potential of a negative “mirror reflection” of λ to the external field). Proof of the theorem follows all the steps of max–min energy method, which are presented in Section 9 for logarithmic potential. First, we need to study continuity properties of the equilibrium energy functional E Ω E Ω F→ −∞ ∞ (6.5) ϕ [F ] = inf ϕ (μ): [ , + ] μ∈F(F ) on F with the Hausdorff metric and establish Theorem . E Ω 6.2 The functional ϕ above is upper semi-continuous. Like the proof of Theorem 3.2 in Section 3, it is convenient to introduce a larger class K of compacts K in Ω satisfying condition s(K) ≤ s, for any K ⊂K.For continuous external field ϕ the functional E Ω K→ −∞ ∞ ϕ : ( , + ) is continuous. Then fields with singular points are approximated by continuous fields. Theorem 6.2 implies that there exist the extremal compact s with E Ω E Ω ∈F (6.6) ϕ [S]= sup ϕ [F ]; S . F ⊂F which leads to the following theorem. Theorem 6.3. The compact S in (6.6) above has Green’s S-property. As in case of logarithmic potential the proof of Theorem 6.3 is based on the fact that equilibrium measure of extremal compact S in (3.7) is a Green’s critical 214 E. A. RAKHMANOV measure associated with external field ϕ andfixedsetA. The part of the proof related to variations may be completely reduced to the case of logarithmic potential. Some details related specifically to Green’s potential are discussed in [30]. 6.5. Best rational approximations to the sum of exponentials on R+. We briefly mention an approximational problem which presents a typical situation m −λ x when conditions of Theorem 6.1 are not satisfied. Let f(x)= cke k where k=1 λk > 0andCk ∈ C;considerρn =minmax |f(x) − r(x)|. We want to find if Γ∈R x∈R+ 1/n lim ρn (if it exists) and to construct a near best approximation. The problem n→∞ is clearly a direct generalization of the case m = 1 discussed in Section 6.3 above. The interpolation approach in Section 6.3 is applied and leads (through orthogonal polynomials as in Section 6.4) to a similar S-problem for Green’s potential in the domain Ω = C ∼ R+ for class F of curves F ⊂ Ω connecting a − i and a + i with a large a>0. The difference with the case m = 1 is that external field ϕ =Rez has to be replaced with the field $ Re z,Rez>0 min λ ϕ(z)= λ = k . λ Re z,Rez<0 max λk This function is not harmonic in Ω (its singularities constitute a whole line — imaginary axis I) and Theorem 6.1 does not apply. One can verify that the max– min method allows us to prove existence of extremal compact F0 ∈Fwhich satisfies E Ω E Ω ∪ ϕ [F0]=max ϕ [F ]. If cap (F0 I)=0thenF0 has the S-property. Most likely F ∈G that F0 ∪ I consists of two points, but the proof of this fact is not known. The problem is open. 6.6. Non-Linear Schr¨odinger. Another example of a Green’s S-problem which does not satisfy conditions of Theorem 6.1 comes from a problem of semi- classical limit for the focusing NLS. The problem may be reduced to the following S-problem for Green’s potential. Let ϕ(z)=Re(az2 + bz + c)+V σ(z) where σ is a positive measure on the interval Δ = [0,iA] of imaginary axis dσ(iy)= η(y)dy, y ∈ [0,A]andη is an analytic (η(y) = 1 would be a typical example). Coefficients a, b, c depend on original variables x, t so that we have a family of quadratic functions; see [29, 30, 28](seealso[41, 62, 63, 11] for other reductions). Let Ω be the upper half-plane and F be a set of all curves F which begins at zero on one side of ∂(Ω ∼ Δ) and go in Ω around Δ to the point zero on the opposite side of ∂(Ω ∼ Δ). Now, the problem is to determine if there is a compact S ∈Fwith Green’s S-property relative to Ω and ϕ. We note that the problem is very similar to the S-problem related to rational approximations for sum of exponentials in Section 6.4 above. The resemblance may apparently be explained by the fact that both problems may be technically formulated using a matrix Riemann–Hilbert problem with 2 × 2 matrices of a similar structure. The two problems in Sections 6.5 and 6.6 have the same property — external field ϕ has a line singularity. We may still consider max–min energy problem and ORTHOGONAL POLYNOMIALS AND S-CURVES 215 find F0 with Eϕ [F0]= supEϕ[F ] F ∈F It is not difficult to prove that F0 (which belongs in general to Ω) does not have a large intersection with R. (Actually F0 ∩ R consists of one or two copies of zero.) However an intersection of F0 and Δ may have a positive capacity and in such cases we actually do not know if original S-problem has a solution or not. Some analysis of the case is presented in [30], but the problem remains open. 6.7. S-problem for vector potentials. For an integer p ≥ 1considerp classes of compacts Fj ⊂Kwith an external field ϕj s defined on Fj ∈Fj , j = 1,...,p. For a fixed vector-compact F =(F1,...,Fp) ∈Fand a vector m = (m1,...,mp) with positive components (total masses on components of vector- compact) we define a family of vector-measures M = M F, m = { μ =(μ1,...,μp):μj /mj ∈M(Fj)} . p For a fixed positive (or non-negative) definite real symmetric matrix A = aij i,j=1 we define associated energy and, subsequently, weighted energy on M P p E ( μ)= aij [μi,μj ]; Eϕ (μ )=E (μ )+2 ϕjdμj # i,j=1 j=1 where [μ, ν]= V ν dμ - mutual energy of μ and ν. ≥ ∩ ∅ Lemma 6.4. If A is non-negative definite and a ij 0 for Fi Fj = then there exist a unique λ ∈ M (equilibrium measure for ϕ, F, m ) Eϕ (λ)= minEϕ( μ). μ∈M (If A is positive definite we may admit small intersections of such components.) For details see original papers [21, 23, 25]) and recent developments in [9, 27]. Let ϕj be harmonic in a neighborhood of Fj .WesaythatS ∈ F has S-property if components of associated vector-equilibrium measure λj satisfy the following conditions: there exist a set e of zero capacity such that for any ζ ∈ supp(λj) ∼ e there exist its neighbourhood D = D(ζ) for which supp (λj ) ∩ D is an analytic arc and, furthermore, we have P ∂Wj ∂Wj (ζ)= (ζ),ζ∈ (supp λ ) ∼ e; W = a V λj + ϕ ∂n ∂n j j i,j j 1 2 i=1 where n1,n2 are opposite normals to supp(λj)atζ. The vector S-existence problem is more complex than the scalar one. We mention briefly some details connected with the vector max–min method. For fixed A,m we define the minimal energy functional Eϕ [F ]onF and, then, corresponding extremal vector-compact S Eϕ [F ] = inf Eϕ( μ)=Eϕ(λ); Eϕ[S ]= supEϕ[F ]. μ∈M(F ) F ∈F Suppose that small variations of S with a vector set A of fixed points belongs to F and also that Si ∩ Sj is a finite set for any two components of S.ThenS has the 216 E. A. RAKHMANOV S-property. This fact is rather simple corollary of variational technique described in Section 4. The existence of maximizing S ∈ F may also be established under essentially same general conditions as in scalar case (e.g., a finite set of singularities of ϕj, a finite fixed set Aj , a finite number of component of Fj ; see Section 9). But the intersections Si ∩ Sj is more difficult to control. At the same time there is comparatively general approach to the vector S- problem from the theory of Riemann surfaces; see first of all J. Nuttall’s papers [45, 46]. In addition, a large number of particular situations are investigated in connection with the applications of vector equilibrium problems to approximation theory, random matrices, statistics, etc.; see [26, 13, 3, 31] and references therein. 7. Strong asymptotics Preceding sections have been devoted to weak asymptotics — zero distribution of orthogonal polynomials Qn with varying weights. In case of fixed (not depending on n) weight it may be characterized by an equilibrium problem without external field. Formulas of strong asymptotics for such polynomials may also be stated in electrostatic terms; then associated equilibrium problems contain an external field 1 of order O( n ). It is convenient, for a fixed n to renormalize the problem and consider measures of total mass n. Then the external field does not depend on n at least in “simply-connected” situations like one in the next subsection (7.1). In general, there is a bounded “effective external field” which depend on n and has several components. In this section we go into some detail in case of orthogonality on one or several intervals of real axis. 7.1. Bernstain-Szeg˝o formulas in electrostatic terms. We consider or- thogonal polynomials Q (x)=xn + ··· on the interval Γ = [−1, 1] n k Qn(x) x w(x) dx =0,k=0, 1,...,n− 1 Γ with a smooth weight w which is positive except, maybe, for a finite set of points. If the Szeg˝o condition log w ∈ L(Γ) is satisfied we have (7.1) Qn(z)=Wn(z)(1+ n(z)) , n(z) → 0,z∈ C ∼ Γ + − + n+ → ∈ (7.2) Qn(x)=Wn (x)+Wn (x)+ n(x), 2 n(x) 0,xΓ 1 − 1 n+ 2 2 4 2 (7.3) Wn(z)=CnD(z) z − 1 z + z − 1 where the Szeg˝o function D(z) ∈ H(C ∼ Γ) is determined by boundary values % % −2 (7.4) %D±(x)% = w(x), (D(∞) > 0),x∈ Γ; n+ 1 (Cn =1/2 2 D(∞) is normalization constant; in what follows we do not write such constant explicitly.) We note that (7.2) is valid apart from zeros of w and endpoints of Γ; see [61, 43, 54, 55] for details. Next, we introduce an external field ϕ associated with the weight w 1 1 (7.5) ϕ(x)= log , or, w(x)=e−2ϕ(x) 2 w(x) and express function Wn in (7.3) above in terms of equilibrium measures in the field ϕ. Note that for a moment we are concerned not with the error of asymptotics Qn ≈ Wn and not with sharp conditions on weight but rather with the form of Wn. ORTHOGONAL POLYNOMIALS AND S-CURVES 217 Let Mn(Γ) be the set of positive Borel measures on Γ with total mass of n and + λ ∈M 1 be the equilibrium measure in the field ϕ. We define n+ 2 1 1 λ = λ+ − δ(1) − δ(−1) ∈M n 4 4 n where δ is the Dirac δ-function. We have (exact or approximate) representations for Wn with z ∈ C ∼ Γandx ∈ Γintermsofλn −Vλn (z) + − (7.6) Wn(z)=e ,Wn (x)+Wn (x)=An(x)cosΦn(x) λn λn + λn where V = V + i V is the complex potential of λn and 1 2 − 1 − 1 (7.7) An(x)=Cn(1 − x ) 4 w 2 (x), Φn(x)=πλn{[x, 1]} = π dλn. x We have exact equality in (7.6) if w>0andw ∈ C1(Γ) so that ϕ ∈ C1(Γ) and n is large enough. Otherwise this representation is approximate which is not significant for applications of Wn in asymptotic formulas since the error of approxi- mation here is generally not large compare to in (7.1). Measure λn is not positive which is not important too since it can be approximated by a positive measure and 1 this can made in many way. One way is to cut two parts of magnitude 4 each from the measure λ+ near the two endpoints of the support. Another way is to define λn the equilibrium measure with normalization Mn(Γ) in the external field 1 2 1 ϕ (x)= log(1/w (x)) where w (x)=(1− x ) 2 w(x), 0 2 0 0 −1/2 (w0 is the trigonometric weight) in these terms we have An(x)=Cnw0 (x)in place of (7.7). A generalization of this way to introduce λn is actually used in multi-connected cases in Sections 7.2 and 7.3 and Section 8 below. Anyway, using electrostatic terms one can define a measure λn ∈Mn(Γ) which plays a role of “a fine, continuous model” for zero distribution of orthogonal poly- nomial Qn. More exactly, zeros of Qn are distributed uniformly with respect to λn (apart from zeros of w and endpoints of the Γ) with errors small with respect to distances between zeros. Equivalently, λn-measure of an interval between two subsequent zeros of Qn is equal to 1 + o(1). Exponential of complex potential of λn may be, then, viewed as a continuous model for the orthogonal polynomial it- self (see (7.7)). In multi-connected cases below we will use generalizations of this definition. Representations in terms of λn are, in particular, useful when ϕ(x)issmooth but it has, say, a finite number of exponential zeros such that the Szeg˝o condition 2 −1/2 1 − x log w(x) ∈ L1(Γ). is not satisfied and the Szeg˝o function D does not exist. Subsequently, Wn in (7.3) does not exist, however, the approximation of this function in (7.6) exists. Asymptotic representation (7.1) (and also (7.2) away from zeros of w)isvalidwith Wn defined in terms of equilibrium measures in (7.6). It seems that this fact was not generally proved yet; see some discussion in [34, 33]. Significant progress has been obtained in the last three decades by the applica- tion of potential-theoretic methods in the investigation of polynomials orthogonal on the whole real axis and more general noncompact sets. Discussion of these results 218 E. A. RAKHMANOV goes beyond the scope of this paper; see, in particular, original papers [50,22,40,51] and books [59, 33, 52]. 7.2. Orthogonality on several intervals. Situation become more compli- cated when from the case of a single interval Γ we pass to the case when Γ is a union of p>1 disjoint intervals Γk =[a2k−1,a2k], k =1,...,p where a1 For each n the collection {ζk,n,sk,n} is uniquely defined by a JIP (7.10) below. Points ζk,n ∈ Ω with sk,n = 1 asymptotically represent zeros of Qn in the gaps between intervals in Γ. We define p −1 2p hn(z)= (z − ζk,n) ,A(z)= (z − ak) . k=1 k=1 With this notation, the Akhiezer–Widom asymptotic formula (see Theorem 6.2 in [65]: see also Secion 14 there) asserts that for z ∈ Ω, x ∈ Γwehave + − + (7.8) Qn(z)=Wn(z)(1+ n(z)) ,Qn(x)=Wn (x)+Wn (x)+ n(x) where n(z) → 0 uniformly on compacts in Ω except for small neighborhoods of 2 points ζk,n (+ n is small with respect to |Wn| in L -norm) and Wn(z)isdefinedby $ ) p−1 h (z) 1 1 (7.9) W (z)=C D(z) n exp n + G(z,∞) − s G(z,ζ ) . n n A1/4(z) 2 2 k k,n k=1 We note that Wn (ζk,n)=0ifsk,n =1andWn (ζk,n) =0if sk,n = −1. Parameters ζk,n, sk,n are defined by the following system of equations p−1 (7.10) sk,nωj (ζk,n)=γj + nωj (∞)[mod2],j=1,...,p− 1 k=1 where πγj is the period of arg D around Γj. System of equations (7.10) is a standard Jacobi Inversion Problem which represent the condition that Wn(z) in (7.9) is single-valued in Ω. ORTHOGONAL POLYNOMIALS AND S-CURVES 219 A version of the formula for the case of varying analytic weight is contained in [16]; a general structure of the function Wn is preserved in case of varying weight. 7.3. Electrostatic Interpretation of the Akhiezer–Widom formula. Here we interpret the function Wn(z) in (7.9) and (7.10) in electrostatic terms as an exponential of a complex equilibrium potential. The construction of corre- sponding measure μn is similar to the one in Section 7.1 but together with the main (continuous) λ-component (we use “trigonometric” version of construction for this component) it contains a finite discrete ν-component reflecting floating zeros of Qn and. In addition, the formula contains an extra set of parameters ζk,n ∈ [a2k,a2k+1] and sk,n = ±1, k =1, 2,...,p− 1. It is, first of all, convenient to represent set of parameters by a single charge σn; the we define charge νn which represent floating zeros of Qn. Thus, we define p−1 1 (7.11) σ = s δ (ζ ) ν = δ (ζ ) n 2 k,n k,n n k,n k=1 sk,n=1 where δ(ζ)isunitmassatζ (in other terms 2σn and νn are divisors on the Double of Ω). Next, for each n ∈ N we define an external field ϕn(x)onΓ 1 1 1 (7.12) ϕ (x)= log − log |A(x)| + V σn (x). n 2 w(x) 4 We note that exp{−2ϕn} presents one of the versions of the “trigonometric weight” for multi-connected case. p −1 1 Finally, let ln = 2 (sk,n +1)bethenumberofpositivesk,n. Then, let λn be k=1 the equilibrium measure on Γ of total mass n − ln in the field ϕn and μn = λn + νn. Theorem 7.1. Suppose that w0(x)=w(x) |A(x)| is positive and smooth. Then for each large enough n ∈ N there exist unique σn of the form (7.11) such that corresponding equilibrium measure λn satisfies condition that λn (Γj ) ∈ Z,j=1,...,p− 1. With this λn we have −Vμn (z) −Vλn (z) Wn(z)=e = e (z − ζk,n) . sk,n=1 The proof is rather straightforward and eventually reduces to comparison of boundary values of log |Wn(x)| and of the potential of μn. Conditions on w0 may be essentially relaxed; then exact representation for Wn in Theorem 7.1 above become approximate. Thus, in the real multi-connected case measure μn = λn + νn representing in a strong sense zeros of orthogonal polynomial is still defined in terms of equilibrium measure with an external field depending on weight function. Compare to simply- connected case the external field is modified by adding a potential of a discrete charge (7.11). The charge is selected to make total mass of λn an integer on each component of the support of weight. It turns out that essentially the same is true for complex orthogonal polynomials in more general situations. A particular case is presented in the next section. 220 E. A. RAKHMANOV 8. Generalized Jacobi polynomials A theorem has been mentioned in Section 2 on zero distribution of Pad´edenom- inators Qn(z) associated with an element f ∈A C ∼ A where A = {a1,...,ap} is a finite set (set of branch points of f). Here we make some remarks on the problem of strong asymptotics for an important particular case of an elements of the form (8.1) below; the case was already briefly discussed in Section 4.2. Let S be Stahl’s continuum for f,thatisf ∈ H C ∼ S and S have minimal n capacity in this class. Polynomials Qn(z)=z + ··· (n ≤ n) — Pade denomiina- tors for f satisfy complex orthogonality relations k k Qn(z)z f(z) dz = Qn(z)z w(z) dz =0,k=0, 1,...,n− 1 S S + − where w(z)=f (z) − f (z)onS (if w ∈ L1(S), otherwise some of the points ak has to be bypassed by small loops in Ω ⊂ C ∼ S). An important fact is that any interval or collection of intervals of real axis has S-property with respect to ϕ = 0 (or any real-symmetric ϕ). Thus, the real case A ⊂ R discussed in Section 7.2 presents an example of our current situation. More exactly, the real case is rather good model of so-called hyper-elliptic situation when Stahl’s continuum S is a union of disjoint Jordan arcs. For hyper-elliptic case with additional condition that f(z) has only quadratic branch points J. Nuttall and R. Singh [48] proved an asymptotic formula similar to the Akhiezer–Widom formula (7.8)–(7.9). We have to add that the Szeg˝o function D(z)=D(z,w) may be defined in complex case as a solution of the boundary value problem D+(ζ)D−(ζ)=1/w(ζ), ζ ∈ S, D(∞) = 0 (also equation (7.10) has to be written in terms of first kind integrals). It was later found out that the method in [48] is similar to the method in an earlier work by N. Akhiezer. Nuttall also stated a general conjecture on the strong asymptotics formula for Pad´e denominators in terms of the solution of a boundary value problem for couple of functions analytic in C ∼ S (somewhat similar boundary value problem is known for hermitian orthogonal polynomials; see [65]). For details see [46]. The proof of a strong asymptotic formula for Qn was not known until recently for hyper-elliptic case without the assumption that branch points are quadratic. General case was essentially completely open. Comparatively general results on strong asymptotics of Pad´e denominators were recently obtained independently in [4] and in [38]. The MRH is used in the first of the two papers and both method MRH and Liouville-Green (WBK) are used in the second one (with the purposes to compare the methods in a situation when both of them may be applied). Here we discuss in some detail results from [38]on strong asymptotics for generalized Jacobi polynomials Qn —Pad´e denominators associated with the function p αk (8.1) f(z)= (z − ak) ,αk ∈ R, αk =0. k=1 For p = 2 this reduces to classical Jacobi polynomials (actually — a particular case of them, because of condition α1 + α2 =0).Thecaseofp = 3 and arbitrary point ak have been investigated by J. Nuttall in [47]; he used Laguerre differential equations for Qn and, then, LG-asymptotics for solutions of this equation. In [38] we combined methods from [47], [36]and[37]. ORTHOGONAL POLYNOMIALS AND S-CURVES 221 Let S be Stahl’s compact for f and V (z)=zp−2 + ··· be associated poly- nomial, so that the complex Green function for S has representation G(z)= #z V (t)/A(t) dt. To simplify the discussion we will assume that S is a contin- a1 uum, that is, all zeros v1,...,vp−2 of V (z) are distinct (this is in a sense a generic case); so, Stahl’s compact is now Chebotarev’s continuum. We also assume that ∈ each aj A is a branch point of f,thatisαj is not an integer; then A and √V do not have common zeros and genus of the Riemann surface R associated with AV is p − 2. Next, we have to introduce special functions for R. They are similar to (but not identical with) special functions of the domain C ∼ [−1, 1] used in Section 7 to construct Akhiezer–Widom formula. It is convenient to introduce them for now as analytic functions in the domain Ω = C ∼ S. We define for z ∈ Ω z dt − Ωk(z)= V (t)/A(t) − ,k=1,...,p 2 a1 t vk — basis of first-kind integrals (analogues of harmonic measures), z dt G(z,ζ)= A(ζ)/V (ζ) V (t)/A(t) − a1 t ζ — third-kind integrals — Green functions for R (here ζ is finite, G(z)=G(z,∞) has been defined above); these functions are multi-valued in Ω but their real parts are single-valued. All the functions above have analytic continuation to R which is in general multi-valued. Like in the real case in Section 7 asymptotic formula for Qn include a set of p,−2 parameters ζk,n ∈ Ω, sk,n = ±1. With hn(z)= (z − ζk,n)wehave(see[38]). k=1 Theorem 8.1. The asymptotic representation Qn(z)=Wn(z)(1 + n(z)) is valid for z ∈ C ∼ S where n(z) is small on compacts in C ∼ S apart from zeros of h (z) and n − hn(z) W (z)=C f(z) 1/2 eHn(z), n n (AV )1/4(z) p−2 p−2 1 1 H (z)= n + G(z) − s G (z,ζ )+ Δ Ω (z): n 2 2 k,n k,n k,n k k=1 k=1 parameters Δ ∈ C, ζ ∈ Ω, s = ±1 are defined by a system of equations k,n k,n k,n dHn = γj (f)mod[Periods],j=1,...,2p − 4 Cj where Cj is a collection of cycles in C ∼ A obtained by projecting a homology basis on R onto the plane (γj depend on f and the choice of cycles). The form of the asymptotic formula in the theorem is still somewhat similar to the form of Akhiezer–Widom formula, but the nature of the situation is more complicated. First, for the same number of branch points we have twice larger genus of associated Riemann surface which is equal to the number of unknown parameters. Second, set of parameters contains two subsets of different nature; accordingly system of equations on parameters (representing conditions on periods 222 E. A. RAKHMANOV of Hn over cycles Cj ) is a combination of a system of equations on periods of first kind integrals with a JIP (this JIP has to be actually stated as a problem on R). Finally, in place of Δk,n one can use another set of p − 2 parameters vk,n which may be called “effective Chebotarev’s centers”. More exactly, straightforward application of LG produces in place of the fixed V (which came from Chebotarev’s p,−2 problem) a polynomial Vn(z)= (z − vk,n) depending on n and, subsequently, k=1 variable Riemann surface Rn. It turns out that Vn converges to V and we can define Δk,n = n (v − vk,n) (those numbers are proved to be bounded). So, there is more than one choice of parameters. Here we have observed one of the differences between methods of LG and MRH. The latter require that we make all the choices for all parameters; the former suggests some choices. Now we briefly outline a construction leading to electrostatic interpretation of the function Wn(z) above. It has some resemblance to what we have seen in Section 7 for the real case but it is based on a more sophisticated equilibrium problem. In place of the equilibrium on a compact set we will have an S-equilibrium problem; existence and characterization of its solution follow by Theorem 3.1. The construction is naturally based on zeros of a variable polynomial Vn and hn as basic parameters. Actually, twice more parameters are involved in analytic description of ˜ situation (see polynomials An, hn below) but the other parameters are determined by basic ones. We fixed n ∈ N and set of parameters ζk,n ∈ Ω=C ∼ S and sk,n = ±1we define charges σn and νn by (7.11). Then we define external fields ϕn (compare (7.12)) and polynomials hn p p−2 α 1 1 k σn ϕn(z)= + log + V (z); hn(z)= (z − ζk,n) . 2 4 |z − αk| k=1 k=1 Let T be a family of continua in C connecting points in A = {a1,...,ap} (if αj < 0 then Γ ∈T goes around ak by a small loop): let ln be the number of positive sj,n. Then Theorem 3.1 implies that there exists a compact Sn ∈T with S-property in the field ϕn relative to measures with total mass n − ln. Thus, Sn is uniquely defined by n ∈ N and set {ζk,n,sk,n,k =1,...,p− 2}. Further, by Theorem 3.1 Sn is a union of 2p − 3 Jordan analytic arcs Sn,j and, maybe, some number of loops Ln,j , all are trajectories of quadratic differential ' ( ! " 2 ˜ 2 2 2 −Rn(z)(dz) ,Rn(z)=Cn An(z)Vn(z)hn(z) / A(z) hn(z) ˜ with polynomials satisfying An → A, Vn → V , hn − hn → 0asn →∞.Further, each Sn,j connect a root of Vn with a root aj,n of An (or, two roots of Vn); for large n we have aj,n ≈ aj where aj is a root of A closest to an,j .Ifforthisj we have αj < 0 then there is a loop Ln,j ∈ S around aj containing aj,n.Inthiscasewe define S˜n,j = Sn,j ∪ Ln,j ; otherwise (if αj >o) there is no loop in Sn around aj and we set S˜n,j = Sn,j and we do same if Sn,j connects two roots of Vn. Finally, for large enough n the compact Sn coincide with support of its equi- librium measure λn. ORTHOGONAL POLYNOMIALS AND S-CURVES 223 Theorem 8.2. There exists a set of parameters {ζk,n,sk,n,k =1,...,p− 2} such that for the equilibrium measure λn of associated S-compact Sn we have λn+σn λn S˜n,j ∈ Z,j=1,...,2p − 4; Wn(z)=exp{−V }. We do not go here into any further details. 9. Proof of the existence theorem 9.1. Extremal energy functional. We consider finite positive Borel mea- sures μ in the extended complex plane C satisfying condition (9.1) log (1 + |x|) dμ(x) < +∞. # μ 1 ∈ −∞ ∞ Then the logarithmic potential V (z)= log |z−x| dμ ( , + ]existsatany z ∈ C and# a.e. finite. Let M be the set of all such measures with finite energy E(μ)= V μ(z) dμ(z) < +∞ (in general E(μ) ∈ (−∞, +∞]). Let K be (here) the family of all compacts K ⊂ C with finite number of (connected) components; at least one of them is not a single point. For K ∈Kwe denote by C∗(K) the set of all real valued functions ϕ which are continuous on K except for a finite set of points e, depending on ϕ;thereare no restrictions on the behavior of ϕ(z)asz → z0 ∈ e.Denote M(K, ϕ)={μ ∈M: ϕ ∈ L1(μ), |μ| =1,Sμ ⊂ K} . We define total energy of μ ∈M(K, ϕ) and equilibrium energy of K ∈K Eϕ(μ)=E(μ)+2 ϕdμ; Eϕ[K] = inf Eϕ(μ). μ∈M(K,ϕ) In Section 9.2, we consider transformation of potentials and energies under linear fractional mappings of C. This would allow us to reduce considerations of compacts in C and measures on them to consideration of compacts and measures in C (and, subsequently, if it is convenient, to the case K ⊂ D1/2 = {z : |z|≤1/2}). In Sections 9.3–9.9, we use K to denote class of compacts in the open plane (we return to spherical compacts in the end of Section 9.10). In Section 9.3 we prove that if Eϕ[K] > −∞ then there exist a unique extremal measure λ = λϕ,K defined by Eϕ(λ)=Eϕ[K]. Section 9.4 contains well known definitions related to balayage needed for further references. Then in Sections 9.5–9.9 we investigate continuity properties of the extremal energy functional Eϕ : K→[−∞, +∞) which is the main subject of Section 9. More exactly, in 9.5 and 9.6 we prove continuity of Eϕ on class of compacts in plane without components of small capacity; we obtain some explicit estimates. In Section 9.7 we prove continuity of Eϕ on class Ks of compacts with bounded number of components in a continuous external field ϕ. In Section 9.8 we extend this result to bounded ϕ. In Section 9.9 we prove that Eϕ[K] is upper semi-continuous under general assumptions. Section 9.10 contains conclusion of the proof of Theorem 3.1. 224 E. A. RAKHMANOV 9.2. Linear fractional transformations of potentials. For a fixed ζ0 ∈ Cζ , c ∈ C consider a mapping of this ζ-plane onto a z-plane and corresponding transformation of measures — measure μ in the z-plane and measure ν in the ζ-plane c c (9.2) z = ,ζ= ζ0 + ; dμ(z)=dν(ζ). ζ − ζ0 z ν If ν ∈Mand V (ζ0) < +∞ then μ defined by (9.2) also belong to M.We have the following relations potentials of μ and ν μ ν ν (9.3) V (z)=V (ζ) − V (ζ0) + log |(ζ − ζ0)/c| ; (9.4) V ν (ζ)=V μ(z)+V μ(0) + log |z| +log|c|. # μ Indeed, making the substitution of x = c/ (t − ζ0)inV (z)=− log |z − x| dμ(x) and using identity c c c(t − ζ) z − x = − = ζ − ζ0 t − ζ0 (t − ζ0)(ζ − ζ0) ν we come to (9.3); (9.4) follows since log |(ζ − ζ0)/c| = − log |z| and V (ζ0)= −V μ(0) − log |c|. Integrating (9.3) with dμ(z)=dν(ζ)wecometo ν (9.5) E(μ)=E(ν) − 2V (ζ0) − log |c|. ∗ Next, let a weight ψ ∈ C (Cζ ) is given in the ζ-plane. We introduce an associated weight in the z-plane by (9.6) ϕ(z)=ψ(ζ) − log |ζ − ζ0| +log|c|. Lemma 9.1. For z and ζ connected by (9.2),measuresdμ(z)=dν(ζ) and external fields ϕ and φ related by (9.6) we have μ ν ν (V + ϕ)(z)=(V + ψ)(ζ) − V (ζ0) Eϕ(μ)=Eψ(ν) + log |c|. Proof. Integration (9.6) with dμ(z)=dν(ζ) yields ν 2 ϕdμ=2 ψdν+2V (ζ0)+2log|c|. By adding (9.5), we obtain a second assertion of Lemma 9.1. The first assertion is obtained by adding up (9.3) and (9.6). ν Let K ∈K, K ⊂ Cζ and ζ0 ∈ K. Then for any ν ∈M(K)wehaveV (ζ0) < + +∞.LetK ∈ Cz be the image of K under linear fractional transformation (9.2). If |c| is small enough then + K ⊂ D1/2 = {z : |z|≤1/2}. Let ψ ∈ C∗(K)andϕ ∈ C∗(K+ ) be related by (9.6) and (9.2). It follows by Lemma 9.1 that there is one-to-one correspondence between measures ν ∈M(K, ψ) and μ ∈M(K,ϕ+ ) defined by dμ(z)=dν(ζ). Furthermore, we have + Eϕ[K]=Eψ[K] − log |c| and if an extremal measure exists for one of the two compacts then it also exists for another one; extremal measures are related by the same equation dμ(z)=dν(ζ). It is not difficult to verify that transformation (9.2) also preserves the S- property, but we are not using this fact. ORTHOGONAL POLYNOMIALS AND S-CURVES 225 9.3. Equilibrium measure. We fix K ∈K, ϕ ∈ C∗(K). Theorem 9.2. If Eϕ[K] > −∞ then there exists a unique measure λ = λϕ,K ∈ M(K, ϕ) with Eϕ(λ)=Eϕ[K] — equilibrium measure for K in the external field ϕ. Proof. We prove the theorem under an additional technical assumption that K = C,sothatK has an exterior point ζ0 in C (we are going to apply the theorem under this restriction; it is, however, not necessary). By the remark in Section 9.2 above, we may assume without loss of generality μ that K ⊂ D1/2. Then for any μ ∈M= M(K, ϕ)wehaveV (z) ≥ 0onK, E(μ) is a norm on M and positive Borel measures on K constitute a complete metric space with this norm. Let μn ∈Mbe a minimizing sequence for Eϕ,thatisEϕ (μn) →E= Eϕ[K]. It follows from the identity μ + σ E(μ − σ)=2E (μ)+2E (σ) − 4E ϕ ϕ ϕ 2 that μn is a Cauchy sequence in metric of energy: E (μn − μm) → 0asn, m →∞. Therefore there is μ on K with finite energy such that E (μn − μ) → 0asn →∞. ∗ As a corollary we have E (μn) →E(μ) and also weak-star convergence μn → μ. Next,wewillprovethatϕ ∈ L1(μ), therefore, μ ∈Mand, subsequently, that μ = λϕ,K . The proof is based on the following assertion stated for the defined above extremal measure μ. It is important to notice that the assertion of Lemma 9.3 is valid for the equilibrium measure μ = λϕ,K of any compact K ⊂ D1/2. Lemma 9.3. For the extremal measure μ above and for constant M defined by (9.7) M =1+inf max ϕ(z)+2γ (K) >0 z∈K where K = K ∼ (e), (e) = {z ∈ K :dist(z,e) < }, we have (9.8) μ({z ∈ K e : ϕ(z) >M})=0. Proof. Since ϕ ∈ C(K ∼ e)thenumberM =maxϕ is finite. If >0issmall K enough then K is not empty and of positive capacity, so that its Robin measure ω has finite energy and the Robin constant γ is finite. For such >0 we define + M = M +2γ, K = {z ∈ K ∼ e : ϕ(z) >M}, and, then, + μ+n = μn − μn| + tnω,tn = μn(K). K # # E + ≤E ≤E ω 2 ω ≤ We have (μn) (μn + tnω) (μn)+2tn V dμn + tn V# dω ω #E (μn)+3tnγ (note that tn ≤ 1andV ≤ γ in C). On the other hand, ϕdμ+n ≤ ϕdμn − Mtn + Mtn. Adding this inequality doubled to the preceding one, we obtain Eϕ (μ+n) ≤Eϕ (μn) − tnϕ < Eϕ (μn) . It follows that μ+n is also a minimizing sequence, that is Eϕ (μ+n) →E. Therefore, it converges in energy (and weak-star) to the same measure μ.SinceK+ is a relative + + open subset of K we have μ(K) ≤ lim μn(K) = 0. Proof of Lemma 9.3 is completed. 226 E. A. RAKHMANOV # It follows by Lemma 9.3 that# ϕ+ dμ < +∞ for ϕ+ #=max{ϕ, 0}.Onthe − − other hand, E > −∞ implies lim ϕ dμn > −∞. Hence, ϕ dμ > −∞,andso ϕ ∈ L1(μ). It remains to prove that Eϕ(μ)=E = lim Eϕ (μn). If it is not so then there n→∞ exist δ>0 and an infinite sequence Λ ⊂ N such that (9.9) ϕdμn ≤ ϕdμ− 3δ, n ∈ Λ. We will show that such an assumption allows us to construct a new sequence of mea- sures μ+n ∈Mwith lim Eϕ (μ+n) < E (actually we can make it −∞) in contradiction with our original assumptions. Let ∈ [0, 1] and ∂e be the boundary of e = {z ∈ K : dis(z,e) > }.Let E = { : μ (∂e)=0} then [0, 1] ∼ E has Lebesgue measure zero (if not then there exist c>0 such that the set { : μ (∂e) ≥ c} has a positive Lebesgue measure; then μ can not be finite). For a fixed ∈ E denote νn = μn| ,ν= μ| ,σn = μn − νn,σ= μ − ν. # # e e We have ϕdσ → ϕdσ. Combined with (9.9), it implies n ϕdνn < ϕdν− 2δ, n ∈ Λ1. # Next, ν = μ| is continuous, ϕ ∈ L (μ), therefore ϕdν → 0as → 0, ∈ E.We e 1 may conclude that there exist a sequence n, n ∈ Λ2, with n ∈ E, n → 0such that ϕdμn < −δ, n ∈ Λ2. en Finally, we introduce a new sequence of measures μ+n =2μn| +(1− tn) μn| ∼ ∈M en K en → →∞ ∈ E + −E → where tn = μn (en ) 0asn ,#n Λ2. It# is clear that (μn) (μn) 0 as n →∞and at the same time ϕdμ+n ≤ ϕdμ − δ, n ∈ Λ2.Fromhere Eϕ (μ+n) ≤Eϕ (μn)−δ is in contradiction with the assumption that μn is minimizing for Eϕ. 9.4. Balayage. Let K ∈Kand Ω be a component of C ∼ K;hereweassume that K ⊂ C and ∞∈Ω. Let g(z,ζ) be the Green function for Ω with a pole at ζ ∈ Ω. Let μ ∈Mbe a measure in Ω, we denote (as in Section 6.1) μ VΩ (z)= g(z,ζ) dμ(ζ) — the Green’s potential of μ.Byμ+ we denote the balayage of μ onto ∂Ω ⊂ K. We note that condition supp μ ⊂ Ω is not necessary; one can assume that balayage does not affect part of μ in the complement to Ω. Lemma 9.4. We have for z ∈ Ω μ μ − μ ∞ (9.10) V (z)=V (z) VΩ (z)+ g(ζ, ) dμ(ζ) (9.11) E (μ+)=E(μ) −EΩ(μ)+2 g(ζ,∞) dμ(ζ) # E Ω μ where (μ)= VΩ (ζ) dζ — Green’s energy of μ. ORTHOGONAL POLYNOMIALS AND S-CURVES 227 # Proof. Denote C = g(ζ,∞) dμ(ζ). The function 1 V (z)= log − g(z,ζ) dμ(ξ)+C |z − ζ| in the right-hand side of (9.10) is harmonic in Ω and it is equal to V μ(z)+C on ∂Ω. As z →∞we have V (z) + log |z|→0 by the symmetry of Green’s function. These are conditions uniquely defining V μ and (9.10) follows. μ − μ ∞ Next, the function V (z) VΩ (z)+g(z, ) is harmonic in Ω; using (9.10) we obtain E + μ + μ − μ + μ − μ + (μ)= V dμ = (V VΩ + C) dμ = (V VΩ + g + C) dμ μ − μ E −EΩ = (V VΩ + g + C) dμ = (μ) (μ)+ gdμ+ gdμ. Equation (9.11) follows and the proof is completed. Let K ∈Kand Ω be a component of C ∼ K. We assume now that infinity is not in Ω; domain may be bounded or not. Again, we consider balayage of a measure μ ∈Monto ∂Ω. Assertions of Lemma 9.4 have to be slightly modified. This leads to the next lemma whose proof is similar to the proof of Lemma 9.4. The situation μ is actually even simpler since VΩ =0on∂Ω. Lemma 9.5. Let μ ∈M, μ+-balayage of μ onto ∂Ω, z ∈ Ω, ∞ ∈ Ω.Then μ μ − μ (9.12) V (z)=V (z) VΩ (z) (9.13) E (μ+)=E(μ) −EΩ(μ) In general, for a compact K ⊂ C the complement Ω = C ∼ K is a union of ∈MC | finite or countable number of components Ωj .Letμ ( ) we define μj = μ Ωj . + Further, let μj be the balayage of μj onto ∂Ωj . We define the balayage μ+ of μ onto K by μ+ = μ+j , j The following lemma is a combination of Lemmas 9.4 and 9.5; note also that μ j μj V (z)=V (z)forz ∈ C ∼ Ωj , ∞ ∈ Ωj. Lemma 9.6. In notations and settings above we have Ωj E (μ+)=E(μ) − E (μj )+2 g(ζ,∞) dμ(ξ) j where summation is taken over all components of C ∼ K and g(z,∞) is the Green function for the component containing ∞ (if there is no such component then the integral term is dropped). 9.5. Lemma on harmonic extension. In this (and the next) section we consider set K of compacts K ⊂ C without small components. More exactly, we assume that for some c>0 capacity of any connected component of K ∈Kis at least c>0andc is same for all compacts (thus, we actually have K = K(c)). Next, let Ω be a domain in C containing K and ϕ ∈ C Ω be a real-valued function (external field) with modulus of continuity ω(δ)inΩ. Finally, let ϕ+(z) be the harmonic extension of ϕ(z)fromK to C;thatisϕ+(z)= ϕ(z)onK, ϕ+(z) is harmonic in any connected component G of C ∼ K (and ϕ+ = ϕ 228 E. A. RAKHMANOV on ∂G). We do not try here to make general settings: actually we need only a bounded harmonic extension of ϕ to some neighborhood of K. Lemma 9.7. Under the assumptions above for any z ∈ C satisfying (9.14) δ =dist(z,K) ≤ dist(z,∂Ω)2; πδ ≤ c; δ ≤ 1 we have √ (9.15) |ϕ(z) − ϕ+(z)|≤4 Mδ1/4 + ω( δ) where M =max|ϕ| and dist(E,F)= min |x − y|. Ω x∈F,y∈F Proof. For a fixed z satisfying (9.14), let x ∈ K a closest to z point on K, that is δ = |z − x|; the interval (z,x) belongs to Ω. Let r ∈ (δ, δ1/2]thenwehave D = {ζ : |ζ − x| (9.16) |ϕ(z) − ϕ+(z)|≤|ϕ(z) − ϕ(x)| + |ϕ+(z) − ϕ(x)|≤ω(r)+|u(z)|. Denote by h(ζ) the harmonic function in D with boundary values h =0on ∂D∩K and h =1ontherestof∂D (so, h is harmonic measure of a part of ∂D in ∂D relative to D). Function u is also harmonic in ∂D and for its boundaty values we have |u(ζ)|≤ω(r)on∂D∩K and |u(ζ)|≤2M on the rest of the boundary. Thus, by maximum principle (9.17) |u(z)|≤2Mh(z)+ω(r)(1 − h(z)) ≤ 2Mh(z)+ω(r). Next, D ∼Dcontains a continuum γ ⊂ F ⊂ K connecting x and ∂D. Define D+ = D ∼ γ: this is an extension of the domain D obtained by removing some components from the complement of the domain in D.Let+h(ζ)beharmonicin D+ with boundary values +h =0onγ and +h = 1 on the rest of the boundary of D+. By the maximum principle we have h(ζ) ≤ +h(ζ), ζ ∈D. Finally, we apply H. Milloux’s theorem (see [18], Chapter VIII, Sec. 4, Theorem 6). The essential part of the theorem asserts that the maximum value for +h(z) at the fixed point z ∈ D+ is reached if γ is a segment of the line from x to ∂D (radius of D)such that z belongs to its continuation. With an explicit expression for corresponding harmonic measures this makes 2 1 − δ/r h(z) ≤ +h(z) ≤ 1 − arcsine < 2 δ/r. π 1+δ/r √ Selecting r = δ and combining this with (9.16) and (9.17), we obtain the assertion of the lemma. We note that this result may be improved by taking min over r ∈ (δ, δ1/2]. It is easy to improve it further to inf over r>δbut we do not need here precise estimates. ORTHOGONAL POLYNOMIALS AND S-CURVES 229 9.6. Continuity of Eϕ for continuous ϕ on compacts without small components. Under assumptions of Section 9.5 above we prove an explicit esti- mate for |Eϕ (K1) −Eϕ (K2)|. More exactly, let Ω be a domain in C, ϕ ∈ C Ω and ω(δ) be the modulus of continuity of ϕ in Ω; assume |ϕ|≤M in Ω. LetK1,K2 be two compacts in Ω; assume that each connected component of each of the two compacts has capacity of at least c>0. Theorem 9.8. Under the conditions above, if δ =dist(K1,K2) ≤ dist (Ki,∂Ω) ; i =1, 2,πδ≤ c then we have √ 1/4 |Eϕ [K1] −Eϕ [K2]|≤4 δ/c + Mδ + ω δ . Theorem 9.8 is essentially a combination of Lemma 9.7 and the following esti- mate for Green’s function which we prove first. Lemma 9.9. Let K ∈Kbe a compact in C and let the capacity of each compo- nent of K be at least c>0.LetΩ be a connected component of C ∼ K with ∞∈Ω and let g(z)=g(z,∞) be the Green function for Ω.Then g(z) ≤ δ/c, δ =dist(z,k),z∈ Ω. Proof. Let x ∈ K be closest to the z ∈ Ω point on K,thatis,|z − x| = δ and F ⊂ K be a component of K with x ∈ F .WedenotebyΩ+ the component + + of C ∼ F with ∞∈Ωandbyg+(z) the Green function for Ω with a pole at ∞. We have Ω ⊂ Ωand+ g(z) ≤ g+(z)inΩ.Itisenoughtoprovethatg+(z) ≤ δ/c; this inequality basically means that the maximum of g+ at a fixed z is attained when F is a segment of length 4c (capacity c) containing z on its continuation with dist(z,F)=δ (compare the end of the proof of Lemma 9.7). A convenient way to obtain a formal proof is to use standard estimates for conformal mappings. Define Φ(z)=eG(z) where g+(z)=ReG(z), G is the complex Green function for Ω+ normalizeg by G(x)=0sothatΦ(x)=1.Asz →∞we have Φ(z)= + z/+c + c0 + c1z + ··· , |+c| = c; Φ is univalent in C ∼ F = Ω and maps this domain onto |ζ| > 1. Point x ∈ ∂Ω+ is accessible, so, the inverse function F =Φ−1 which maps conformally |ζ| > 1ontoΩ+ is (non-tangent) continuous at y =1isζ-plane. We have F (ζ)=c (ζ + α0 + α1/ζ + ···)at∞. Now by Theorem 1 in [18](Chapter IV) we have |F (ζ)−F (y)|≥c|ζ −y|2/ζ. From here with ζ =Φ(z)andy =Φ(x)=1 we obtain 1 δ |Φ(z) − 1|2 ≤ |z − x|·|Φ(z)|≤ |Φ(z)|. c c On the other hand, we have |Φ(z) − 1|≥|Φ(z)|−1=eg (z) − 1; from here and the above inequality 2 g+(z)2 ≤ eg (z)/2 − e−g (z)/2 ≤ δ/c and Lemma 9.9 is proven. Proof of Theorem 9.8. Let λ be the equilibrium measure for K2 and let μ be the balayage of λ onto K . By Lemma 9.5 1 Eϕ [K1] ≤Eϕ(μ) ≤Eϕ(λ)+2 (g + ϕ+ − ϕ) dλ 230 E. A. RAKHMANOV where g = g(z,∞)-Green’s function for component of C ∼ K1 containing ∞, ϕ+ is C ∼ + harmonic in each component of K1, ϕ = ϕ on K1. By Lemma 9.9 we √ have g(z) ≤ δ/c on supp(λ) and by Lemma 9.6 we have |ϕ+ − ϕ|≤4 Mδ1/4 +ω δ √ 1/4 on supp λ. Altogether it makes E [K1] ≤E[K2]+4 Mδ + δ/c + ω δ . Since conditions on K1,K2 are symmetric, Lemma 9.8 follows. 9.7. Continuity of Eϕ for continuous φ on compacts with bounded number of components. Theorem 9.8 asserts that for a continuous external field ϕ in C, the functional of equilibrium energy Eϕ[K] is continuous on the space of compacts K ⊂ C with cap(F ) ≥ c>0 for each connected component F of each compact K. We did not assume that the number of components is bounded. Now we drop the condition cap(F ) ≥ c>0oncomponentsofK and introduce a restriction on the number of components. For a fixed s ∈ N we recall notation Ks for the set of all compacts K ∈ C with at most s components and positive capacity. We will prove that for continuous ϕ the functional of weighted equilibrium energy is continuous on Ks with Hausdorff metric. We do not include explicit estimates. Theorem 9.10. For a continuous ϕ(z):C → R the functional Eϕ[K]:Ks → (−∞, +∞) is continuous. Proof. We need to show that for any K0 ∈Ks and any >0 there exist δ>0 such that for any K ∈Ks we have (9.18) |Eϕ[K] −Eϕ [K0]| < for δH (K, K0) <δ First we consider the case when K0 does not have components consisting of a single point. Then there is a positive minimum for capacities of components: cap (F0) ≥ c0 > 0 for any component F0 of K. There is also a positive minimum 2δ0 for distances between different connected components. Then for any δ<δ0 the set (K0)δ consists of s0 ≤ s domains, each containing one connected component of K0; s0 is the number of components of K0. For any K ∈Ks, δH (K0,K) <δand for any connected component F0 ⊂ K0, the domain (F0)δ contains a compact part F ⊂ K such that F0 ⊂ (F )δ.IfF ⊂ K associated with each component of K0 is a continuum, then the assertions which we want to prove follow by Theorem 9.8. m In general, we have F = Fj where m ≤ s and Fj are connected components j=1 of K.Form>1, we will describe a procedure of connecting components Fj of F + + + + into a continuum F such that F ⊂ F , δH (F, F ) and cap(F ∼ F ) are small. Then we will do the same with each subset — union of components of K associated with a component of K ; we obtain a compact K+ ∈K with K ⊂ K+ , δ K, K+ is small 0 s H and cap K+ ∼ K is small. Then we prove that the last condition implies that % % % + % %Eϕ[K] −Eϕ[K]% is small, which will conclude the first part of the proof. In the process of constructing F+, we also use the following simple observations. The equilibrium energy functional is decreasing, that is, K1 ⊂ K2 implies Eϕ [K2] ≤ E E ≤E ϕ [K1]. We have, therefore, ϕ[(K0)δ] ϕ[K]. Both compacts K0 and (K0)δ E →E → satisfy conditions of Theorem 9.8, which implies ϕ (K0)δ ϕ [K0]asδ 0. ORTHOGONAL POLYNOMIALS AND S-CURVES 231 ⊂ On the other hand, from δH (K, K0) <δwe obtain K (K0)δ and, further, for any >0 (9.19) Eϕ[K] ≥Eϕ [K0] − , δ ≤ δ( ). Thus, we need only to prove that (9.20) Eϕ[K] ≤Eϕ [K0]+ , δ ≤ δ( ) to obtain (9.18). When proving (9.20), we may drop subsets of K and pass to smaller compacts K ⊂ K; indeed, if (9.20) is proved for K = K then it is proved for K since Eϕ[K] ≤Eϕ [K ]. Now we are ready to describe the construction of continuum F+.Webeginwith m F = Fj , m ≤ s, Fj -continua, δH (F0,F) <δ. j=1 Suppose first that dist (F1,F ∼ F1) ≥ 2δ. Since continuum F0 belongs to ∪ ∼ (F1)δ (F F1)δ and sets are disjoint, we conclude that F0 belongs to one of these two sets. Then we can drop another one and pass to F ⊂ F which still satisfies δH (F ,F0) <δand has at least one connected component less than F . Then we continue the process with F in place of F (see remark above). In the opposite case dist (F1,F ∼ F1) < 2δ there exist a point z in the intersec- ∩ ∼ ∈ ∈ ∼ | − |≤ tion (F1)δ (F F1)δ.Letx1 F1 and x2 F F1 are such that z x1 δ and + |z − x2|≤δ.LetΣ1 be the union of segments [z1,x1]and[z1,x2]andF1 = F ∪ Σ1. + The number of components of F1 is at most m − 1; Σ1 ⊂ Dδ (z1) — disc of radius + + δ and δH (F, F1) ≤ δ.Thus,δH (F0, F1) < 2δ. After we repeat this operation ≤ m − 1 times, we construct a continuum F+ = m−1 F ∪ ΣF where ΣF is contained in a union of ≤ m − 1 discs with radii ≤ 2 δ. + m−1 We have δH F0, F ≤ 2 δ. We can apply the described procedure to the subset + F ⊂ K associated with each component of K0. It gives us a compact K = K ∪ ΣK s + ≤ s ⊂ ≤ s with δH K0, K 2 δ,ΣK Drj (zj ), rj 2 δ. j=1 + Compact constructed above K hasasmanycomponentsasK0 has and com- + s ponents of K are 2 δ close to components of K0; this implies that capacities of + components of K are ≥ c0/2 > 0 for small enough δ. Thus, by Theorem 9.8 we have for any >0 % % % + % (9.21) %Eϕ K −Eϕ [K0]% ≤ /2,δ≤ δ( ) (δ( ) here and in (9.19)–(9.20) depends also on s, R =max|z|, M =max|ϕ|). z∈K0 We finish the proof using the following lemma. + Lemma 9.11. Let ϕ ∈ C DR , K = K∪Σ ⊂ DR be compacts, c =cap(K) > 0. Then for any >0 there exist η( ) such that + + Eϕ[K] ≤Eϕ[K] ≤Eϕ[K]+ /2 for cap(Σ) <η( ). (η( ) depends on ϕ, c, R, but does not depend on K, Σ.) Proof of this lemma is presented at the end of the section. Now we complete the proof of the theorem 9.10. s We have Σ ⊂ Dr (zj ) ⊂ DR (for, say, R =max|z| +1 and δ ≤ 1), j ∈ j=1 z K0 s rj ≤ r =2δ. Itisclearthatcap(Σ)→ 0asδ → 0 under these conditions 232 E. A. RAKHMANOV 1/s uniformly for {zj } (actually,wehavecap(Σ)≤ CRδ with an absolute constant C). Then (9.18) is a combination of (9.21) and Lemma 9.11. To be precise, we mention again that in general our operations with K satisfying + δH (K0,K) <δyields not K = K ∪ ΣK with the same condition, but K ⊂ K ∪ ΣK + with δH (K0,K ) <δ(this K ,notK satisfies the conditions of Theorem 9.8). Then we proceed with Lemma 9.11 and one-sided inequalities (9.19) and (9.20) as indicated in the comment above related to those inequalities. It remains to consider the case when K0 ∈Ks has degenerated components. Let K1 = K0 + L0 where K0 is a union of all non-degenerated continua as above and L0 is a finite number of points in C ∼ K0. We have Eϕ [K1]=Eϕ [K0]. Any compact K ∈Ks with δH (K1,K) <δwith for small enough δ>0 will have representation K = K + L where δH (L, L0) <δ, δH (K ,K0) <δ. We have in general cap(L) > 0 but L is contained in a finite union of discs of radius = δ,sothatcap(L)issmallasδ →∞. By Lemma 9.11, it follows that |Eϕ [K ] −Eϕ[K]| is also small for small δ. Thus, the situation is reduced to the case of compact K0 without point-components. This concludes the proof of the theorem. Proof of Lemma 9.11. We start with a few elementary observations. ∈M ⊂ 1 For a measure μ , supp μ DR, R> 2 ,wehave (9.22) V μ(z) ≥−|μ| log(2R), |z|≤R (9.23) E(μ) ≥−|μ|2 log(2R) where |μ| = μ(C). Equation (9.22) follows by inequality log(1/|z − x|) ≥−log 2R for z,x ∈ DR; (9.23) follows by (9.22). For measures λ = μ + ν in DR we have (9.24) E(μ) ≤E(λ)+2|λ||ν| log(2R). # Indeed, by (9.22) we have E(λ)=E(μ + ν)=E(μ)+ V 2μ+ν dν ≥E(μ) −|ν||2μ + ν| log(2R) and (9.24) follows. Next we will show that for a unit measure λ ∈Min DR and a compact set Σ ⊂ DR we have E(λ)+2log(2R) (9.25) λ(Σ)2 ≤ γ(Σ) where γ(Σ) is a Robin constant of Σ. Denote also by ωΣ the Robin measure for Σ. | − E | | ≥E Let μ = λ Σ, ν = λ μ.Wehave (μ/ μ ) (ωΣ)=γ(Σ). On the other hand, E(μ/|μ|)=E(μ)/|μ|2 = E(μ)/λ(Σ)2.Thusλ(Σ)2 ≤E(μ)/γ(Σ). Combined with (9.24) it gives (9.25). Next, the left inequality in Lemma 9.11 is a corollary of monotonicity of Eϕ[K]. To prove the one on the right, we consider the equilibrium measure λ for K+ = K∪Σ. | − | | − We again set μ = λ Σ, ν = λ μ and, further, t = μ = λ(Σ), r =1/(1 t). Since rν is a unit measure on K we have 2 2 Eϕ[K] ≤Eϕ(rν)=r Eϕ(ν)+2r ϕdν = r E(λ − μ)+2r ϕd(λ − μ) (9.26) ≤ r2(E(λ)+2t log(2R)) + 2r ϕdλ− 2r ϕdμ. ORTHOGONAL POLYNOMIALS AND S-CURVES 233 (To obtain the last inequality, we used (9.24) with μ and ν interchanged). We may assume that t ≤ 1/2(see9.27below)thenwehave r − 1 ≤ 2t, r2 − 1 ≤ 6t. + Besides, with ω = ω we have Eϕ(λ) ≤Eϕ(ω) ≤ γ K +2M and further E(λ) ≤ # K + Eϕ(λ) − 2 ϕdλ≤ γ K +4M. Taking into account those inequalities, we obtain from (9.26) + + Eϕ[K] ≤Eϕ K + t 6γ K +32M + 8 log(2R) . At the same time, it follows by (9.25) that γ K+ +4M + 2 log(2R) (9.27) t2 = λ(Σ)2 ≤ . γ(Σ) If cap(Σ) is small, then γ(Σ) is large uniformly over all Σ ⊂ DR.Atthesame time, cap K+ ≥ c>0 implies that γ K+ ≤ 1/ log 1 so that t is small and c + Eϕ[K] −Eϕ K is also small uniformly for K,Σ⊂ DR,cap(K) ≥ c>0. Lemma 9.11 is proven. ∗ ∗ ∗ 9.8. Continuity of Eϕ for bounded ϕ ∈ C . We denote by the C = C (C) set of real-valued functions ϕ continuous in C ∼ e for some finite set e ⊂ C (de- pending on ϕ). Here we prove that the continuity of Eϕ[K]onKs which is asserted by Theorem 9.10 for ϕ ∈ C(C) remains valid for ϕ ∈ C∗ under the additional assumption of boundedness of ϕ. ∗ Lemma 9.12. Let ϕ ∈ C and |ϕ|≤M in C ∼ e.ThenEϕ[K]:Ks → (−∞, ∞) is continuous. In order to prove Lemma 9.12, we need the following two auxiliary assertions. First of them is a folklore Lemma 9.13. For a fixed R ≥ 1 there is a constant C = C(R) such that for any two compacts E,F ∈ DR we have cap(E ∪ F ) ≤ C(cap(E)+cap(F )). Lemma 9.14. For fixed s ∈ N, R>0 and any >0,thereexistδ( )= δ( , s, R) > 0 such that for any two compacts E,F ∈ DR we have | cap(E) − cap(F )| < for δH (E,F) <δ( ). To prove Lemma 9.14 it is enough to note the following. If both capacities are small the assertion of the lemma is trivial. If one of them is bounded away from zero then the assertion is a particular case of Theorem 9.10 with ϕ =0.Now,we turn to the proof of Lemma 9.12 + ∈K E + For a fixed K s we need to show that functional ϕ is continuous at K.Let γ = γ K+ (it is a finite number and c =cap K+ = e−γ > 0), let R =max|z| +1. z∈K P Next, let e = {a1,...,ap}; define Dr = Dr (aj )-union of discs Dr (aj )of j=1 radius r centered at aj .Forr ∈ (0, 1], we define the function ϕr(z) be continuous in C,harmonicinDr,andϕr = ϕ in C ∼Dr;thus,ϕr(z) is a regularization of ϕ near singular points. 234 E. A. RAKHMANOV Let λ be the equilibrium measure for K+ in the field ϕ and λr be the equilibrium measure of the same compact in the field ϕr.Wehave E + ≤E r E r r E r − r ϕ K ϕ (λ )= (λ )+2 ϕdλ = ϕr (λ )+2 (ϕ ϕr) dλ . From here (note also that we can interchange ϕ and ϕ ) % % r % % %E + −E + % ≤ r D ϕ K ϕr K 4Mλ ( r) Further, by (9.25) with Σ = Dr we have as r → 0 r + (9.28) λ Dr ≤ γ(K)+4M + 2 log(2R) /γ Dr → 0. Combined with the inequality above, it makes % % % % %E + −E + % → → (9.29) ϕ(K) ϕr (K) 0asr 0. The same arguments show that this relation is also valid for any compact K ∈Ks + with δH (K, K) <δif δ is small enough |E −E |→ → (9.30) ϕ(K) ϕr (K) 0asr 0 and, moreover, this limit is uniform with respect to K with the indicated properties. Indeed, according to (9.28) the left-hand side in (9.29) is bounded by 4M(γ(K)+4M + 2 log(2R))/γ Dr which is uniformly small as γ Dr is large since γ(K) is bounded by Lemma 9.13. Now, assertion of Lemma 9.12 follows by (9.29) and (9.30) in combination with Theorem 9.10. ∗ 9.9. Semi-continuity of Eϕ for ϕ ∈ C . First, we prove that the assertion of Lemma 9.12 remains valid even if we remove the the assumption that ϕ is bounded from above. ∗ Lemma 9.15. Let ϕ ∈ C and ϕ(x) ≥−M in C ∼ e.ThenEϕ is continuous on Ks. Proof. Again, for a fixed K˜ ∈Ks we need to show that Eϕ is continuous at −γ K˜ .Wehavec =cap(K˜ )=e > 0. By Lemma 9.14 there is δ0 > 0 such that for any K ∈Ks, δH (K, K˜) <δ≤ δ0 ≤ 1wehavecap(K) ≥ c/2orγ(K) ≤ γ +ln2. We also set R =1+max|z| then K ∈ DR for any K ∈Kwith δH (K, K˜) ≤ δ0. z∈K˜ For any such compact K we may apply (9.7) and (9.8) in Lemma 9.3 to the equilibrium measure λ = λϕ,K in place of μ. We need to take into account that formulas (9.7) and (9.8) were derived for the case K ∈ D1/2. To reduce the current case to the one in Lemma 9.3 we make substitution ζ = z/2R to map a neigh- ˜ bourhood of K onto class of compacts in D1/2. Then we come to the following: for M =inf max ϕ(z)+2γ(K ∼D)+2log2R +1 0 ∈ ∼D >0 z K R we have λ ({z ∈ K : ϕ(z) >M0}) = 0. The only term depending of K in the formula for M0 is γ(K ∼D). By Lemmas 9.13 and 9.14 the last expression is bounded for all K ⊂ (K˜ ) if is small enough. Thus, we can find M such that λ({ϕ ≥ δ0 1 M }∩K) = 0 for any K ⊂ (K˜ ) . 1 δ0 ORTHOGONAL POLYNOMIALS AND S-CURVES 235 { } E E ∈ ˜ Thus, forϕ ˜(z)=min ϕ(z),M1 we have ϕ[K]= ϕ˜[K] for any K (K)δ0 . Butϕ ˜ is bounded from above and from below and Lemma 9.15 follows by Lemma 9.12. ∗ Now, we finish the proof of Theorem 3.2. Let ϕ ∈ C . Define ϕn =max{ϕ, −n}. E E ≤E ϕn is continuous by Lemma 9.15. On the other hand we have ϕn [K] ϕn+1 [K] ∈K E for K s.Thus ϕn is monotone decreasing sequence of continuous functions on Ks and its limit is upper semi-continuous. Theorem 3.2 is proven. 9.10. Proof of Lemma 3.5 and Theorem 3.4. Proof of Lemma 3.5 is based on the following general fact: first variation of equilibrium energy of a compact coincide with the first variation of energy of equilibrium measure. We reproduce here the proof from [49]wherecaseϕ = 0 was considered. Presence of external field does not essentially affect arguments used in the proof (taking into account Remark 4.1 we can assume that functions h in variations are harmonic in a neighbourhood of K; then the external energy term is analytic in t). Let K ∈Ks be a compact in C, ϕ is harmonic in Ω ∼ e where K ⊂ Ωande is a finite set. We will consider a variation z → zt = z + th(z) associated with a smooth function h with h(z)=0forz ∈ (e)δ with some small enough δ>0. Let λ = λϕ,K t t and μt be the equilibrium measure for K = {z ,z ∈ K} in the field ϕ. For small enough t>0 mapping z → zt is one-to-one, so that there exist the inverse mapping −t −t t t −t −t −t z = z − th(z)+o(t), that is (z ) =(z ) = z.Letμ =(μt) ∈M(K). We define d = D E (λ), d(t)=D E (μ ) (see (4.2) and (4.3)). h ϕ h ϕ t E Kt ≤E λt = E (λ)+td + O(t2)=E [K]+td + O(t2) ϕ ϕ ϕ ϕ −t 2 t 2 Eϕ[K] ≤Eϕ μ = Eϕ (μt) − td(t)+O(t )=Eϕ[K ] − td(t)+O(t ). It is not difficult to verify that constants in all O(t2) are explicit and uniform for small enough t.Fromhere 2 t 2 td(t)+O(t ) ≤Eϕ K −Eϕ[K] ≤ td + O(t ) It implies, in particular, that μt and λ are close in energy metric as t → 0, so that d(t) → d as t → 0 and, subsequently that 1 t 1 t (9.31) lim Eϕ K −Eϕ[K] = lim Ep λ −Eϕ(λ) . t→0+ t t→0+ t It is important to observe now that the assertion (9.31) remains valid if K is a compact in the extended complex plane under the assumption that the point at infinity is included in the set e of singular points of ϕ. Then the condition h(z)=0 for z ∈ (e)δ means that the variation leaves a neighbourhood of infinity fixed and all the arguments above remain valid. Now we can complete proofs. We return to a class T of compact satisfying conditions of Theorem 3.1. There exist a maximizing sequence Γn. Next we select a subsequence from {Γn} which converges in spherical δH -metric to S. By condition (iv) there is a disc D such that Γn ∼ D ∈T and, therefore, converges to S ∼ D.SinceEϕ [Γn ∼ D] ≥Eϕ [Γn] sequence Γn ∼ D is also maximizing. Since sup Eϕ[Γ] is finite the sequence of Γ corresponding equilibrium measures λn converges in energy to measure λ.By Lemma 9.1 we can reduce situation to the case where Γn and S belong,say,to the disc D 1 . By Theorem 3.2 we have Eϕ[S]=supEϕ[Γ], λ is equilibrium measure 2 Γ for S. 236 E. A. RAKHMANOV It follows that for any smooth h vanishing in (e)δ with a positive δ we have 1 t lim (Eϕ[S ] −Eϕ[S]) = 0. From here and (9.31) we obtain that DhEϕ(λ)=0 t→0+ t for any such h. This implies that λ is (A, ϕ)-critical and, therefore, its weighted potential has S-property. Acknowledgement. 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MR0239059 (39:418) Department of Mathematics & Statistics, University of South Florida, Tampa, Florida 33620-5700 E-mail address: [email protected] Contemporary Mathematics Volume 578, 2012 http://dx.doi.org/10.1090/conm/578/11471 Fast decreasing and orthogonal polynomials Vilmos Totik Abstract. This paper reviews some aspects of fast decreasing polynomials and some of their recent use in the theory of orthogonal polynomials. 1. Fast decreasing polynomials Fast decreasing, or pin polynomials have been used in various situations. They imitate the ”Dirac delta” best among polynomials of a given degree. They are an indispensable tool to localize results and to create well localized ”partitions of unity” consisting of polynomials of a given degree. We use the setup for them as was done in [5], from where the results of this section are taken. Let Φ be an even function on [−1, 1], increasing on [0, 1], and −Φ(x) suppose that Φ(0) ≤ 0. Consider e , and our aim is to find polynomials Pn of a given degree ≤ n such that −Φ(x) (1.1) Pn(0) = 1, |Pn(x)|≤e ,x∈ [−1, 1]. Let nΦ = n be the minimal degree for which this is possible. The following theorem gives an explicitly computable bound for this minimal degree. Theorem 1.1. (Ivanov–Totik [5]) 1 N ≤ n ≤ 12N , 6 Φ Φ Φ where Φ(x) NΦ =2 sup 2 Φ−1(0)≤x<Φ−1(1) x 1/2 Φ(x) Φ(x) + dx +sup +1. 2 − − Φ−1(1) x 1/2≤x<1 log(1 x) Here Φ−1(t)=sup{u Φ(u) ≤ t} is the generalized inverse. It can be shown that each term can be dominant in NΦ, but, in the most important cases, the second term gives the order of nΦ. 2000 Mathematics Subject Classification. Primary 42C05. Supported by ERC grant No. 267055. c 2012 American Mathematical Society 241 242 VILMOS TOTIK Theorem 1.1 is universal in the sense that the function Φ may have parameters, in particular it may include the degree n of the polynomial. In applications mostly the following two types of decrease is used. Let ϕ, ϕ(0) ≤ 0, be an even function on [−1, 1]. Assume that ϕ is increasing on [0, 1] and there is a constant M0 such that ϕ(2x) ≤ M0ϕ(x) for all 0 Corollary 1.2. There are Pn of degree at most n =1, 2,... with −cnϕ(x) Pn(0) = 1, |Pn(x)|≤Ce ,x∈ [−1, 1], (where c, C > 0 are independent of n),ifandonlyif 1 ϕ(u) ∞ 2 du < . 0 u On the other hand, if ψ, ψ(0) ≤ 0, is an even function on R for which ψ is increasing on [0, ∞)andthereisaconstantM0 such that ψ(2x) ≤ M0ψ(x) for all x>0, then we have Corollary 1.3. There are Pn of degree at most n =1, 2,... with −cψ(nx) Pn(0) = 1, |Pn(x)|≤Ce ,x∈ [−1, 1], (where c, C > 0 are independent of n),ifandonlyif ∞ ψ(u) ∞ 2 du < . −∞ 1+u { }∞ In Corollary 1.2 the decrease of Pn(x) n=1 is exponential at every x =0.In Corollary 1.3 this decrease is somewhat worse, but the polynomials Pn start to get small very close to 0 (e−cψ(nx) start having effect from |x|∼1/n). As concrete examples consider Example 1.4. −cn|x|α Pn(0) = 1, |Pn(x)|≤Ce ,x∈ [−1, 1], with some Pn of degree at most n =1, 2,... (andwithsomec, C > 0) is possible precisely for α>1. Example 1.5. −c(n|x|)β (1.2) Pn(0) = 1, |Pn(x)|≤Ce ,x∈ [−1, 1], with some Pn of degree at most n =1, 2,... (andwithsomec, C > 0) is possible precisely for β<1. In particular, −cn|x| Pn(0) = 1, |Pn(x)|≤Ce ,x∈ [−1, 1], is NOT possible for polynomials of degree at most n. It easily follows from Theorem 1.1 that to have this decrease one needs deg(Pn) ≥ cn log n. FAST DECREASING AND ORTHOGONAL POLYNOMIALS 243 2. Quasi-uniform zero spacing of orthogonal polynomials n Let μ be a Borel-measure on [−1, 1] with infinite support, let pn(x)=γnx +··· denote the orthonormal polynomials with respect to μ, and let xn,1 γj dμ(x)=h(x) |x − xj | dx, γj > −1,h>0 continuous, are doubling. On the other hand, if dμ(x)=|x|γ for −1 ≤ x<0, and dμ(x)=|x|δ for 0 244 VILMOS TOTIK the zero spacing is quasi-uniform regardless that the weight is much stronger around −1/2 than around 1/2. Of course, finer spacing will distinguish such differences in theweight(seee.g.[23]). Y. Last and B. Simon [8] had the first results on local zero spacing if only local information is used on the weight. They proved that 1. if dμ(x)=w(x)dx and for some q>0 A|x − Z|q ≤ w(x) ≤ B|x − Z|q in a neighborhood of a point Z,then(withaC independently of n) C |x(1)(Z) − x(−1)(Z)|≤ n n n (−1) (1) where xn (Z) ≤ Z ≤ xn (Z) are the zeros enclosing Z,and 2. if w is bounded away from 0 and ∞ on I,then(withac independently of n) c |x(1)(y) − x(−1)(y)|≥ n n n inside I. This was extended to locally doubling measures by T. Varga: Theorem 2.2. (Varga [24]) If μ is doubling on an interval I,then 1 x − x ∼ n,k+1 n,k n locally uniformly inside I. The endpoint version of this is: Theorem 2.3. (Totik-Varga [19]) If μ is doubling on I =[a, b] and μ((a − ε, a)) = 0,then √ x − a 1 x − x ∼ n,k+1 + n,k+1 n,k n n2 locally uniformly for xn,k ∈ [a, b − ε]. So this holds around local endpoints of the support (i.e. at which, for some ε>0, μ((a − ε, a)) = 0 but μ((a, a + ε)) =0). Zero spacing is connected to the measure via Christoffel functions and the Markov inequalities. So to see how fast decreasing polynomials enter the picture in connection with zero spacing we have to discuss Christoffel functions. 3. Christoffel functions Recall the definition of the n-th Christoffel function associated with a measure μ: 2 λn(x) = inf |Pn| dμ, Pn(x)=1 where the infimum is taken for all polynomials of degree at most n taking the value 1 at the point x. It is well known (and easily comes from the minimality property (2.3)) that − n 1 2 λn(x)= |pk(μ, x)| . k=0 FAST DECREASING AND ORTHOGONAL POLYNOMIALS 245 Their importance lies in the fact that Christoffel functions, unlike the orthogonal polynomials, are monotone in the measure μ (as well as in their index n). Hence, they are much easier to handle than the orthogonal polynomials themselves. For them the following rough asymptotics was proved. Theorem 3.1. (Mastroianni-Totik [11]) If μ is doubling, then for x ∈ [−1, 1] λn(x) ∼ μ([x − Δn(x),x+Δn(x)]) uniformly in n and x ∈ [−1, 1]. Recall also the Cotes numbers: λn,k = λn(xn,k), which appear in Gaussian quadrature 1 n fdμ ∼ λn,kf(xn,k). − 1 k=1 For them Theorems 2.1 and 3.1 easily give Theorem 3.2. (Mastroianni-Totik [11]) If μ is doubling, then for all n and 1 ≤ k (3.4) λn(x) ≤ Cμ([x − 1/n, x +1/n]),x∈ [−3/4, 3/4]. Indeed, then from the Markov inequality (3.3) and from (3.4), we have (3.5) μ([x ,x ]) ≤ λ + λ n,k n,k+1 n,k n,k+1 ≤ C μ([xn,k − 1/n, xn,k +1/n]) + μ([xn,k+1 − 1/n, xn,k+1 +1/n]) . 246 VILMOS TOTIK We may assume xn,k+1 − xn,k > 4/n, since otherwise there is nothing to prove. Then we set I =[xn,k − 1/n, xn,k+1 +1/n], E1 =[xn,k − 1/n, xn,k +1/n]and E2 =[xn,k+1 − 1/n, xn,k+1 +1/n]. Using the doubling property (2.1) and the bound (3.5), we get μ(I) ≤ Lμ([xn,k,xn,k+1]) ≤ CL(μ(E1)+μ(E2)) . Now it can be shown that the doubling property implies that with some K and r>0 |E | r μ(E ) ≤ K 1 μ(I), 1 |I| |E | r μ(E ) ≤ K 2 μ(I). 2 |I| Consequently, the preceding inequalities yield K 1 ≤ 2CL (|E | + |E |)r |I|r 1 2 i.e. 2 x − x < |I|≤(2CLK)1/r . n,k+1 n,k n Thus, it is enough to prove (3.4), and this is where fast decreasing polynomials enter the picture. Since we want to prove a local result like (3.4) from the local assumption that μ is doubling in a neighborhood I of x, we may assume that x =0∈ I =[−a, a] and supp(μ) ⊂ [−1, 1]. Take fast decreasing polynomials Pn of degree at most n such that −c(n|x|)1/2 Pn(0) = 1, |Pn(x)|≤Ce ,x∈ [−1, 1] (see (1.2)). On [2k/n, 2k+1/n] ⊂ [−a, a]wehave k/2 |Pn(x)|≤C exp(−c2 ), and at the same time, by the doubling property of μ on [−a, a], we have μ([2k/n, 2k+1/n]) ≤ μ([2k/n, (2k +2k+1)/2n]) ≤ L2μ([2k−1/n, 2k/n]) ≤ ···≤L2kμ([0, 1/n]). Hence, 1 2 k/2 2k −c(na/2)1/2 |Pn| dμ ≤ C exp(−c2 )L μ([0, 1/n]) + e μ([−1, 1]). 0 2k/n≤a Here the sum is convergent, and it is easy to see that the doubling property implies μ([0, 1/n]) ≥ (c/ns) with some s, so the preceding inequality gives 1 2 |Pn| dμ ≤ Cμ([0, 1/n]). 0 A similar estimate holds for the integral over [−1, 0], and this verifies (3.4). FAST DECREASING AND ORTHOGONAL POLYNOMIALS 247 4. Nonsymmetric fast decreasing polynomials Symmetric fast decreasing polynomials that we have considered up to now, are not enough to prove this way the endpoint case, namely Theorem 2.3. The problem is not in the requirement that the bound e−Φ(x) is a symmetric function; indeed, if Φ is not even, then one can consider instead of it the symmetric Φ(x)+Φ(−x) which is at least as large as Φ(x) (minus an irrelevant constant). However, so far we have requested that (1.1) should hold on [−1, 1], i.e. there is a control on Pn on a relatively long interval to the right and to the left from the peaking point 0. If the left-interval where one needs to control Pn is considerably shorter (like in the endpoint case), then one can get faster decrease. Theorem 4.1. (Totik-Varga [19]) For β<1 there are C, c > 0 such that for all x0 ∈ [0, 1/2] there are polynomials Qn(t) of degree at most n =1, 2,... such that Q (x )=1, n 0 ⎛ ⎞ β cn|t − x | |Q (t)|≤C exp ⎝− 0√ ⎠ ,t∈ [0, 1]. n | − | t x0 + x0 √ Note that here the denominator is ∼ x0 on [0, 2x0], which, for such x,results in a large positive factor in the exponent when compared to what we have in the symmetric case. For example, in the extreme case when x0 = 0 we get: there are Pn of degree at most n such that Pn(0) = 1 and −cnxγ |Pn(x)|≤Ce ,x∈ [0, 1], precisely if γ>1/2. Compare this with Example 1.4 according to which in the symmetric case −nxβ |Pn(x)|≤Ce ,x∈ [−1, 1], is possible precisely if β>1. Now the upper estimate in Theorem 4.1 goes through the Markov inequalities and the estimate of the Christoffel function: (4.1) λn(x) ≤ Cμ([x − δn(x),x+ δn(x)]) where √ x − a 1 δ (x)= n,k+1 + n n n2 exactly as in the proof in the preceding section; and (4.1) follows from Theorem 4.1 as the analogous result (3.4) followed from (1.2). 5. Fast decreasing polynomials on the complex plane For a long time fast decreasing polynomials and their applications were re- stricted to the real line. Recently it has turned out that they also exist on more general sets on the complex plane and they play a vital role in some questions related to orthogonal polynomials. Let K ⊂ C be a compact subset of the complex plane and Z ∈ K.Ofcourse, if Z lies in the interior of K (or in the interior of one of the connected components of its complement) then, by the maximum modulus principle, there are no fast decreasing polynomials on K that peak at Z. The situation is different if Z lies on the so called outer boundary of K, defined as the boundary ∂Ω of the unbounded component of the complement C \ K. 248 VILMOS TOTIK Theorem 5.1. (Totik [17], [20]) Let Z ∈ ∂Ω be a point on the outer boundary of K. Assume there is a disk in Ω that contains Z on its boundary. Then, for γ<1, there are a c>0 and polynomials Qn of degree at most n =1, 2,... such that Qn(Z)=1, |Qn(z)|≤1 for z ∈ K and (i): type I: −c(n|z−Z|)γ |Qn(z)|≤Ce ,z∈ K, (ii): type II: −cn|z−Z|1/γ |Qn(z)|≤Ce ,z∈ K. These two types of decrease are the analogues of Examples 1.4 and 1.5. Here, exactly as on the real line, γ = 1 is not possible. We also mention, that the assumption that there is a disk in Ω containing Z on its boundary is very natural; in fact, it cannot be replaced e.g. by the assumption that there is a cone/wedge in the complement of opening <πwith vertex at Z. In the next sections we shall give applications of these complex fast decreasing polynomials. 6. Christoffel functions on a system of Jordan curves Recall that a Jordan curve is the homeomorphic image of the unit circle C1, while a Jordan arc is the homeomorphic image of the interval [0, 1]. Let E be a finite system of smooth (C2) Jordan curves and let μ be a Borel- measure on E. We assume that there are infinitely many points in the support of μ. The definition of the Christoffel functions is the same: 2 λn(μ, z) = inf |Pn| dμ, Pn(z)=1 and we have again that if pn(μ, z) are the orthonormal polynomials, then n 2 1/λn(μ, z)= |pk(μ, z)| . 0 To describe the asymptotic behavior of λn on E, we need the concept of equi- librium measures. The equilibrium measure μE of E minimizes the logarithmic energy 1 log dν(z)dν(t) |z − t| among all Borel-measures ν supported on E having total mass 1. We shall also define the equilibrium density ωE as the density (Radon-Nikodym derivative) of the equilibrium measure with respect to arc length measure s on E: dμE = ωE ds. The same concepts can be defined for arcs, and even for more general sets. For example, √ 1 ω[−1,1](x)= , π 1 − x2 while for a circle/disk of radius r we have ωE ≡ 1/2πr, i.e. in this case the equilibrium measure lies on the bounding circle and it has constant density there (the constant coming from the normalization to have total mass 1). With these notions we can state FAST DECREASING AND ORTHOGONAL POLYNOMIALS 249 Theorem 6.1. (Totik [17], [20]) Let E be a finite family of C2 Jordan curves, and assume that μ is a Borel-measure on E for which log μ ∈ L1(s),whereμ is the Radon-Nikodym derivative of μ with respect to arc measure s. Then at every Lebesgue-point z0 of μ and log μ μ(z ) lim nλ (μ, z )= 0 . →∞ n 0 n ωE(z0) Recall, we say that z0 ∈ γ is a Lebesgue-point (with respect to s)forthe integrable function w if 1 lim |w(ζ) − w(z0)|ds(ζ)=0, → s(J) 0 s(J) J where the limit is taken for subarcs J of E that contain z0, the arc length s(J) of which tends to 0. Also, if dμ = wds + dμs is the decomposition of μ into its absolutely continuous and singular part with respect to s,thenz0 is a Lebesgue- point for μ if it is Lebesgue-point for w and μ (J) lim s =0. s(J)→0 s(J) There is a local version of Theorem 6.1, where the smoothness of E and the 1 Szeg˝o condition μ ∈ L (s) is assumed only in a neighborhood of z0 (see [17], [20]). The theorem is also true when some of the curves are replaced by arcs, but the proof for the arc case is completely different (the polynomial inverse image approach to be discussed below cannot be used; an arc has no interior, it cannot be exhausted by lemniscates), and is, again, based heavily on complex fast decreasing polynomials. T N w0 z0 E Figure 1. Creating lemniscates Sketch of the proof of Theorem 6.1 There are two distinctively different parts: the continuous case has been dealt with in [17], while the case of general Lebesgue-points in [20]. 250 VILMOS TOTIK Part I. Continuous case: μ is absolutely continuous and μ = w is continuous and positive at z0. In this case we use a polynomial inverse mapping (see [21]). a): The result is known for the unit circle C1 (Szeg˝o). ∗ −1 b): Go over to a lemniscate E = TN (C1)whereTN is an appropriate fixed polynomial (see Figure 1). ∗ −1 c): Approximate E by a leminscate E = TN (C1) containing z0 (see Figure 2). Here, in part c), fast decreasing polynomials of type II (see Theorem 5.1) are used in a very essential way. For the approximation in part c) one also needs an extension of Hilbert’s lemniscate theorem: Suppose that Γ is another system of C2 Jordan curves consisting of the same number of components as E such that each component of Γ lies in the corresponding component of E with the exception of the point z0, where the two (system of) curves touch each other and have different ∗ −1 curvatures. Then there is a lemniscate E = TN (C1) consisting of the same number of component and which separates E and Γ (and of course touch both at z0). z0 E* E Figure 2. Approximating E by a lemniscate Part II. Reduction to the continuous case. μ (z0) 1): Set dν = ds on the component of E that contains z0, and let ν = μ ωE (z0) on other components. The density of this ν is just constant on the component of E which contains z0,soforthisν Part I applies at z0. 2): Show that λn(ν, z0)=(1+o(1))λn(μ, z0). Here, and in many similar questions, the main problem is how to control the size of the optimal polynomials in 2 λn(ν, z) = inf |Pn| dν. Pn(z)=1 This problem is handled by the following inequality. FAST DECREASING AND ORTHOGONAL POLYNOMIALS 251 Theorem 6.2. (Totik [20]) Let γ be a C2 Jordan curve and w ≥ 0 ameasur- 1 able function on γ with w, log w ∈ L (s).Ifz0 ∈ γ is a Lebesgue-point for log w, then there is a constant M such that for any polynomials Pn of degree at most n =1, 2,... and for any z ∈ γ (or for z lying inside γ) √ 2 M n|z−z0| 2 (6.1) |Pn(z)| ≤ Me n |Pn| wds. γ This is a fairly non-trivial estimate, for example nothing like this is true outside γ: n Example 6.3. Let γ be the unit circle, w ≡ 1, Pn(z)=z , z0 =1.Then,for z>1, |P (z)|2 = z2n =(1+(z − 1))2n ≥ en(z−1), n √ and here the right hand side is far from being ≤ MeM n|z−1| n. The crucial idea is to combine Theorem 6.2 with fast decreasing polynomials of type I (see Theorem 5.1): Qεn(z0)=1, 2/3 −(εn|z−z0|) |Qεn(z)|≤Ce ,z∈ E. − | − | 2/3 Now√ no matter how small ε>0 is, the factor e (εn z z0 ) kills the factor M n|z−ζ0| e in (6.2), so the product PnQεn is bounded on γ and is very small away from z0. At the same time, it has almost the same degree at Pn,andwecan use these as test polynomials to estimate the Christoffel functions for the measure ν (or μ) in Part II.1) above. Using the Lebesgue-point property and these test polynomials, it is relatively easy to verify Part II.2). 7. Universality Let w be an integrable weight function on some compact set Σ ⊂ R, and let pk be the orthonormal polynomials associated with w. Form the so called reproducing kernel n Kn(x, y)= pk(x)pk(y). k=0 A form of universality of random matrix theory/statistical physics at a point x claims a b Kn x + ,x+ − w(x)Kn(x,x) w(x)Kn(x,x) → sin π(a b) Kn(x, x) π(a − b) as n →∞. This was proved under analyticity of w in various settings by different authors (see e.g. Pastur [12], Deift, Kriecherbauer, McLaughlin, Venakides and Zhou, [2] or Kuijlaars and Vanlessen [6], [7]). D. S. Lubinsky [10] proved it under mere continuity: if Σ = [−1, 1] and w>0 is continuous in (−1, 1), then universality is true at every x ∈ (−1, 1). Actually, he proved universality at an x ∈ (−1, 1) if w(x)dx ∈ Reg and w>0 is continuous at x.HereReg is the class of measures μ for which 1/n lim inf λn(μ, x) ≥ 1 at every point x of the support with the exception of a set of zero logarithmic capacity. This is a weak global condition on the measure, and it says that for most points x in the support the value |Pn(x)| of polynomials is not exponentially larger 252 VILMOS TOTIK 2 than their L (μ)-norm Pn L2(μ).See[15] for various reformulations of regularity and for different regularity criteria. Extension of Lubinsky’s universality to general support and to almost every- where convergence (under Szeg˝o condition) was done by Simon [13], Findley [4] and Totik [18]. Lubinsky had a second, complex analytic approach to universality, which was abstracted by Avila, Last and Simon [1]: universality is true at a point x0 ∈ S if (i): 1 ω (x ) lim K (μ; x + a/n, x + a/n)= Σ 0 →∞ n 0 0 n n w(x0) uniformly in a ∈ [−A, A] for any fixed A, (ii): there is a C>0 such that for any A>0, |z|≤A and for sufficiently large n ≥ nA 1 K (x + z/n,x + z/n) ≤ CeC|z|,z∈ C. n n 0 0 Since 1/λn(μ, x)=Kn(x, x), property (i) is basically the asymptotics for Christoffel functions discussed before (with the small change x0 → x0 + a/n, called by Simon the “Lubinsky wiggle”, see Remark 3 on p. 225 of [14]). On the other hand, (ii) is not that easy to verify at a given non-continuity point. Now (ii) follows from the inequality (6.1) with the use of fast decreasing polynomials of type II (see Theorem 5.1) at every point which is a Lebesgue-point for w and log w.Thisway one gets Theorem 7.1. (Totik [22]) Let μ ∈ Reg and dμ(x)=w(x)dx on an interval 1 I with log w ∈ L (I). Then universality is true at every x0 ∈ I which is a Lebesgue- point for both w and log w. In particular, it is true a.e. That universality is true almost everywhere under a local Szeg˝o condition was proved in [18] by a totally different method (using polynomial inverse images). It should be noticed that these two absolutely different approaches (namely in [18]and Theorem 7.1) need the same assumption, namely local Szeg˝o condition w ∈ L1(I). It is an open problem if this Szeg˝o condition can be replaced by something weaker (like w>0 a.e. in I). 8. The Levin-Lubinsky fine zero spacing theorem Let again w be an integrable weight, but now assume that its support is [−1, 1], and let xn,k be the zeros of the associated orthogonal polynomials pn(μ, x). The following remarkable result was proved as a consequence of Lubinsky’s universality theorem. Theorem 8.1. (Levin-Lubinsky [9]) If w>0 is continuous on (−1, 1),then − 2 1 xn,k x − x =(1+o(1)) n,k+1 n,k n uniformly for xn,k ∈ [−1+ε, 1 − ε]. Actually, Levin and Lubinsky proved more, namely that the same is true if it is only assumed that w(x)dx ∈ Reg, w is continuous and positive at a point X, and |xn,k − X| = O(1/n). FAST DECREASING AND ORTHOGONAL POLYNOMIALS 253 The extension to more general support was given independently by Simon and Totik: Theorem 8.2. (Simon [13],Totik[18]) Let μ be a measure on the real line with compact support S in the Reg class. Assume also that dμ(x)=w(x)dx, log w ∈ L1(I) on some interval I. Then at every X ∈ I which is a Lebesgue-point for w and log w, we have 1+o(1) (8.1) xn,k+1 − xn,k = , |X − xn,k| = O(1/n), nπωS(X) where ωS denotes the equilibrium density of S with respect to linear Lebesgue- measure. In [13] the continuity and positivity of w wasused,andin[18]asomewhat less precise result (as regards where (8.1) holds) was verified. 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MR2019609 (2004j:42025) 24. T. Varga, Uniform spacing of zeros of orthogonal polynomials for locally doubling measures, (manuscript). Department of Mathematics and Statistics, University of South Florida, 4202 E. Fowler Ave, PHY 114, Tampa, Florida 33620-5700 and Bolyai Institute, Analysis Re- search Group of the Hungarian Academy os Sciences, University of Szeged, Szeged, Aradi v. tere 1, 6720, Hungary E-mail address: [email protected] CONM 578 rhgnlPlnmas pca ucin,adApplications and Functions, Special Polynomials, Orthogonal This volume contains the proceedings of the 11th International Symposium on Orthogo- nal Polynomials, Special Functions, and their Applications, held August 29–September 2, 2011, at the Universidad Carlos III de Madrid in Leganes,´ Spain. The papers cover asymptotic properties of polynomials on curves of the complex plane, universality behavior of sequences of orthogonal polynomials for large classes of measures and its application in random matrix theory, the Riemann–Hilbert approach in the study of Pade´ approximation and asymptotics of orthogonal polynomials, quantum walks and CMV matrices, spectral modifications of linear functionals and their effect on the associated or- thogonal polynomials, bivariate orthogonal polynomials, and optimal Riesz and logarith- mic energy distribution of points. The methods used include potential theory, boundary values of analytic functions, Riemann–Hilbert analysis, and the steepest descent method. • reúe l,Editors al., et Arvesú ISBN 978-0-8218-6896-6 9 780821 868966 CONM/578 AMS