Orthogona l Polynomial s on the Uni t Circl e Par t 1: Classica l Theor y This page intentionally left blank http://dx.doi.org/10.1090/coll054.1

America n Mathematica l Societ y Colloquiu m Publication s Volum e 54 , Par t 1

Orthogona l Polynomial s on the Uni t Circl e Par t 1: Classica l Theor y

Barr y Simo n

America n Mathematica l Societ y Providence , Rhod e Islan d Editorial Boar d Susan J . Priedlander , Chai r Yuri Mani n Peter Sarna k

2000 Mathematics Subject Classification. Primar y 42C05 , 05E35 , 34L99 ; Secondary 47B35 , 30C85 , 30D55 , 42A10 .

For additiona l informatio n an d update s o n thi s book , visi t www.ams.org/bookpages/coll-54

Library o f Congres s Cataloging-in-Publicatio n Dat a Simon, Barry , 1946 - Orthogonal polynomial s o n the uni t circl e / Barr y Simon . p. cm. — (American Mathematical Societ y colloquium publications, ISS N 0065-925 8 ; v. 54 ) Contents: pt . 1 . Classica l theor y Includes bibliographica l reference s an d index . ISBN 0-8218-3446- 0 (par t 1 : alk . paper)—ISB N 0-8218-3675- 7 (par t 2 : alk . paper ) 1. Orthogona l polynomials . I . Title . II . Colloquiu m publication s (America n Mathematica l Society) ; v. 54 . QA404.5.S45 200 4 515'.55—dc22 200404621 9

AMS softcover ISB N 978-0-8218-4863- 0 (par t 1) ; 978-0-8218-4864-7 (par t 2) .

Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them , ar e permitted t o mak e fai r us e o f the material , suc h a s to cop y a chapter fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provided th e customar y acknowledgmen t o f the sourc e i s given . Republication, systemati c copying , or multiple reproduction o f any material i n this publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc h permission shoul d be addressed to the Acquisitions Department, America n Mathematica l Society , 201 Charle s Street , Providence , Rhod e Islan d 02904-2294 , USA . Request s ca n als o b e mad e b y e-mail to reprint-permissionQams.org . © 200 5 by the America n Mathematica l Society . Al l rights reserved . Reprinted b y the America n Mathematica l Society , 2009 . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o the Unite d State s Government . Printed i n the Unite d State s o f America . @ Th e pape r use d i n this boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . Visit th e AM S hom e pag e a t http://www.ams.org / 10 9 8 7 6 5 4 3 2 1 1 4 1 3 1 2 1 1 1 0 0 9 To my grandchildren an d thei r parent s This page intentionally left blank Contents

Preface t o Part 1 x i Notation xvi i Chapter 1 Th e Basic s 1 1.1 Introductio n 1 1.2 Orthogona l Polynomial s o n the Rea l Lin e 1 1 1.3 Caratheodor y an d Schu r Function s 2 5 1.4 A n Introduction t o Operato r an d Spectra l Theor y 4 0 1.5 Verbhmsk y Coefficient s an d the Szeg o Recurrence 5 5 1.6 Example s o f OPUC 7 1 1.7 Zero s and the First Proo f o f Verblunsky's Theore m 9 0 Chapter 2 Szego' s Theorem 10 9 2.1 Toeplit z Determinant s an d Verbhmsk y Coefficient s 10 9 2.2 Extrema l Properties , the Christofie l Functions , an d th e Christoffel-Darboux Formul a 11 7 2.3 Entrop y Semicontinuit y an d the Firs t Proo f o f Szego' s Theorem 13 6 2.4 Th e Szeg o Function 14 3 2.5 Szego' s Theorem Usin g the Poisso n Kernel 15 1 2.6 Khrushchev' s Proo f o f Szego's Theorem 15 6 2.7 Consequence s o f Szego' s Theorem 15 9 2.8 A Higher-Order Szeg o Theorem 17 2 2.9 Th e Relativ e Szeg o Function 17 8 2.10 Totik' s Workshop 18 4 2.11 Ries z Products an d Khrushchev' s Worksho p 18 9 2.12 Th e Workshop o f Denisov an d Kupi n 19 7 2.13 Matrix-Value d Measure s 20 6 Chapter 3 Tool s fo r Geronimus ' Theore m 21 7 3.1 Verblunsky' s Viewpoint : Proof s o f Verblunsky's an d Geronimus' Theorem s 21 7 3.2 Secon d Kin d Polynomial s 22 2 3.3 K W Pair s 23 9 3.4 Coefficien t Strippin g an d Associate d Polynomial s 24 5

Chapter 4 Matri x Representation s 25 1 4.1 Th e GG T Representatio n 25 1 4.2 Th e CM V Representatio n 26 2 4.3 Spectra l Consequence s o f the CM V Representation 27 4 4.4 Th e Resolven t o f the CM V Matrix 28 7 viii CONTENT S

4.5 Ran k Tw o Perturbations an d Decouplin g o f CM V Matrices 29 3 Chapter 5 Baxter' s Theore m 30 1 5.1 Wiener-Hop f Factorizatio n an d th e Inverse s o f Finite Toeplit z Matrice s 30 1 5.2 Baxter' s Proo f 31 3 Chapter 6 Th e Stron g Szeg o Theorem 31 9 6.1 Th e Ibragimo v an d Golinskii-Ibragimo v Theorem s 31 9 6.2 Th e Borodin-Okounko v Formul a 33 3 6.3 Representation s o f U(n) an d th e Bump-Diaconi s Proo f 34 6 6.4 Toeplit z Determinant s a s the Statistica l Mechanic s o f Coulomb Gase s and Johansson' s Proo f 35 2 6.5 Th e Combinatoria l Approac h an d Kac' s Proo f 36 8 6.6 A Second Loo k at Ibragimov' s Theore m 37 6 Chapter 7 Verblunsk y Coefficient s Wit h Rapi d Deca y 38 1 7.1 Th e Rate o f Exponential Deca y an d a Theorem o f Nevai-Totik 38 1 7.2 Detaile d Asymptotic s o f the Verblunsky Coefficient s 38 7 Chapter 8 Th e Densit y o f Zeros 39 1 8.1 Th e Densit y o f Zero s Measure vi a Potential Theor y 39 1 8.2 Th e Densit y o f Zero s Measure via the CM V Matrix 40 3 8.3 Rotatio n Number s 41 0 8.4 A Gallery o f Zero s 41 2 Bibliography 42 5 Author Inde x 45 7 Subject Inde x 46 3 Preface t o Part 2 x i Notation xii i Chapter 9 Rakhmanov' s Theore m an d Relate d Issue s 46 7 9.1 Rakhmanov' s Theore m vi a Polynomial Ratio s 46 7 9.2 Khrushchev' s Proo f o f Rakhmanov's Theore m 47 5 9.3 Furthe r Aspect s o f Khrushchev's Theor y 48 5 9.4 Introductio n t o MNT Theor y 49 3 9.5 Rati o Asymptotic s 50 3 9.6 Poincare' s Theore m an d Ratio Asymptotics 51 2 9.7 Wea k Asymptotic Measure s 52 1 9.8 Rati o Asymptotic s fo r Varyin g Measure s 53 0 9.9 Rakhmanov' s Theore m o n an Ar c 53 5 9.10 Wea k Limit s an d Relativ e Szeg o Asymptotics 53 8 Chapter 1 0 Technique s o f Spectral Analysi s 54 5 10.1 Aronszajn-Donoghu e Theor y 54 5 10.2 Spectra l Averagin g an d the Simon-Wolf f Criterio n 55 1 10.3 Th e Gordon-de l Rio-Makarov-Simo n Theore m 55 8 10.4 Th e Grou p U(l,l) 56 4 CONTENTS i x

10.5 Lyapuno v Exponent s an d th e Growt h o f Norms i n U(l, 1) 58 1 10.5A Appendix: Subshift s 60 0 10.6 Furstenberg' s Theore m an d Rando m Matri x Product s Prom U(l, 1 ) 60 6 10.7 Th e Transfe r Matri x Approac h to L 1 Verblunsk y Coefficient s 61 7 10.8 Th e Jitomirskaya-Las t Inequalitie s 63 1 10.9 Criteri a fo r A.C . Spectrum 63 9 10.10 Dependenc e o n the Tai l 64 8 10.11 Kotan i Theor y 65 2 10.12 Priife r Variable s 66 4 10.13 Modifyin g th e Measure : Insertin g Eigenvalue s an d Rational Functio n Multiplicatio n 67 3 10.14 Deca y o f CMV Resolvents an d Eigenfunction s 68 5 10.15 Countin g Eigenvalue s i n Gaps: Th e Birman-Schwinge r Principl e 69 0 10.16 Stochasti c Verblunsk y Coefficient s 70 1 Chapter 1 1 Periodi c Verblunsk y Coefficient s 70 9 11.1 Th e Discriminan t 71 0 11.2 Floque t Theor y 71 9 11.3 Calculatio n o f the Weigh t 72 4 11.4 A n Overvie w o f the Invers e Spectra l Problem 73 0 11.5 Th e Orthogona l Polynomial s Associate d t o Dirichle t Dat a 74 2 11.6 Wal l Polynomials an d the Determinatio n o f Discriminants 74 8 11.7 Abel' s Theore m an d the Invers e Spectra l Proble m 75 3 11.8 Almos t Periodi c Isospectral Tor i 78 3 11.9 Quadrati c Irrationalitie s 78 8 11.10 Independenc e o f Spectral Invariant s an d Isospectra l Tor i 79 9 11.11 Isospectra l Flow s 80 1 11.12 Bound s o n the Green' s Functio n 80 8 11.13 Genericit y Result s 81 1 11.14 Consequence s o f Many Close d Gap s 81 2 Chapter 1 2 Spectra l Analysi s o f Specifi c Classe s of Verblunsky Coefficient s 81 7 12.1 Perturbation s o f Bounded Variatio n 81 7 12.2 Perturbation s o f Periodic Verblunsk y Coefficient s 82 6 12.3 Naboko' s Workshop : Dens e Point Spectru m i n the Szeg o Clas s 82 9 12.4 Generi c Singula r Continuou s Spectru m 83 4 12.5 Spars e Verblunsk y Coefficient s 83 8 12.6 Rando m Verblunsk y Coefficient s 84 5 12.7 Decayin g Random Verblunsk y Coefficient s 84 7 12.8 Subshift s 85 5 12.9 Hig h Barrier s 86 3 Chapter 1 3 Th e Connectio n to Jacob i Matrices 87 1 13.1 Th e Szeg o Mapping an d Geronimu s Relation s 87 1 13.2 CM V Matrices an d th e Geronimu s Relation s 88 1 13.3 Szego' s Theorem fo r OPRL : A First Loo k 88 9 13.4 Th e Denisov-Rakhmano v Theore m 89 2 13.5 Th e Damanik-Killi p Theore m 89 6 x CONTENT S

13.6 Th e Geronimo-Cas e Equation s 90 3 13.7 Jacob i Matrice s With Exponentiall y Decayin g Coefficient s 91 2 13.8 Th e P 2 Su m Rul e and Application s 92 0 13.9 Szego' s Theorem fo r OPRL : A Third Loo k 93 7 Appendix A Reader' s Guide : Topic s and Formula e 94 5 A.l What' s Don e Where 94 5 A. Schu r function s 94 5 B. Toeplit z matrice s an d determinant s 94 5 C. Szego' s theorem 94 6 D. Aleksandrov familie s 94 6 E. Zero s o f OPUC 94 6 F. Densit y o f zeros 94 6 G. CM V matrices 94 7 H. Periodic Verblunsky coefficient s 94 7 I. Stochasti c Verblunsk y coefficient s 94 7 J. Transfe r matrice s 94 8 K. Asymptotic s o f orthogonal polynomial s 94 8 A.2 Formula e 94 8 A. Basic objects 94 8 B. Recursio n 95 0 C. Bernstein-Szego approximatio n 95 5 D. Additional O P formula e 95 6 E. Additiona l Wal l polynomial formula e 95 6 F. Matri x representations 95 6 G. Aleksandro v familie s 95 9 H. Rotation o f measure 96 0 I. Sieve d polynomials 96 0 J. Toeplit z determinants (se e also K) 96 1 K. Szego' s theory 96 1 L. Additional transfe r matri x formula e 96 4 M. Periodic Verblunsk y coefficient s 96 6 N. Connectio n t o Jacob i matrice s 96 7 Appendix B Perspective s 97 1 B.l OPR L vs . OPUC 97 1 B.2 OPU C Analog s o f the ra-function 97 3 Appendix C Twelv e Grea t Paper s 97 5 Appendix D Conjecture s an d Ope n Question s 98 1 D.l Relate d t o Extending Szego' s Theorem 98 1 D.2 Relate d to Periodi c Verblunsk y Coefficient s 98 1 D.3 Spectra l Theor y Conjecture s 98 2 Bibliography 98 3 Author Inde x 103 1 Subject Inde x 1039 Preface t o Par t 1

For an overvie w o f the subject o f these volumes , se e Section 1.1 .

Here i s how this boo k cam e to be . I'v e worke d i n part o n the spectra l theor y of Schrodinge r operator s fo r al l o f m y career . I n th e lat e 1970s , i t becam e clear , especially fo r those o f us trying to understand localizatio n i n random systems, tha t it wa s technicall y easie r t o stud y th e discret e Schrodinge r operator , whic h i n on e dimension read s

(hu)n = izn+i + w n_i + v nun (1 ) In abou t 1985 , I me t Moura d Ismai l wh o urge d m e t o loo k a t th e literatur e o n orthogonal polynomial s — and I should hav e paid mor e attention to hi s advice. At leas t I did loo k a little a t orthogona l polynomial s o n the rea l lin e (OPRL ) and learne d that theor y wa s essentially the stud y o f operators o f the for m

{Ju)n = a n+iwn+i + bn+iUn + a nun-i (2 ) where n = 0,1,2,... an d (oi , c&2,...) and (bi , &2 » • • •) are real with a n > 0 for n > 1 and a o = 0 . Th e discret e Schrodinge r cas e i s a n = 1 . One thin g I realize d earl y i s that th e Jacob i matri x (2 ) i s the natura l aren a for invers e theor y an d tha t i t seeme d difficul t t o clarif y wha t spectra l measure s corresponded to an = 1 with bn arbitrary. A s inverse problems became an important component o f my research, I worked o n what wa s essentially OPRL , eve n goin g s o far as writing a long review article [974 ] on moment problems that coul d be thought of as a primer on OPRL. But I didn't systematicall y loo k at literature fro m the O P community. My firs t rea l exposur e t o orthogona l polynomial s o n th e uni t circl e (OPUC ) came i n connectio n wit h m y rol e a s a n edito r o f Communication s i n Mathemati - cal Physics (CMP) , i n particular, wit h a submission o f Golinski i an d Neva i [464] . My sectio n i n CM P ha d accepte d a n earlie r pape r o n th e subjec t b y Geronim o and Johnso n [398] , but a t the time I hadn't pai d clos e attention to it. I personall y knew both those authors and the paper seemed to be studying some kind o f general difference equation , s o I sen t i t t o a n appropriat e refere e an d followe d th e recom - mendation. Th e Golinskii-Neva i pape r wa s different . I knew o f Nevai's reputatio n but didn't know either author. Th e paper was clearly about orthogonal polynomials and I wa s reluctant, give n CMP' s perpetua l pag e crunch , t o ope n u p th e journa l to a subjec t tha t I worrie d migh t b e disconnecte d t o eithe r physic s o r th e area s that w e normally cover . S o I spen t som e tim e carefull y readin g th e introduction , thinking abou t th e issues , and skimmin g the rest. I realized this paper wa s one o n spectral theor y relate d t o man y paper s i n CMP , s o I wa s comfortabl e sendin g i t out t o a referee an d followin g a positive recommendation .

xi xii PREFACE T O PAR T 1

The next aspects o f the story involve my own interests and those o f my research group. A major them e o f our wor k in the 1990 s concerned the spectral behavior o f discrete Schrodinge r operator s (1 ) an d th e continuum analo g i n one dimension : (Hu)(x) = -u"(x) + V(x)u(x) (3 ) It ha s bee n know n sinc e the 1930 s that V G L1 o r \V(x)\ 1 (equivalently, a n = 1 and b n el o r \b n\ < C(l + \n\)~ fo r the discret e case) i s a natural dividin g line. I f V G L1, the H give n by (3 ) has purely absolutel y continuous spectru m a t positiv e energie s an d scatterin g states . In m y ow n work, i t becam e clea r a = | wa s als o a natural dividin g line . Thi s appeared firs t i n wor k I did o n decaying rando m potential s [965] ; there wa s dens e point spectrum i f a < \ an d subsequent wor k [265, 268, 649] showed a.c. spectrum if a > \. I saw it agai n when I proved that Bair e generic potentials decayin g slowe r than n-1/2 hav e purely singula r continuou s spectru m [969] . I the n emphasize d tha t th e cas e \ > a > \ wa s open , an d Kiselev , i n hi s thesis don e unde r m e [637] , an d the n Christ-Kiselev-Remlin g [201 ] filled i n th e region showin g there wa s alway s a.c . spectrum i n this regime . Kiselev-Last-Simo n [639], motivate d b y thi s work , conjecture d tha t th e righ t borderlin e conditio n fo r a.c. spectru m wa s L 2. Rowa n Killip , a graduat e studen t I wa s supervisin g a t the time , go t intereste d i n thi s questio n an d h e aske d Perc y Deif t abou t i t whil e Percy wa s visiting . Perc y notice d tha t th e Kd V su m rul e involve d L 2 precisely , suggesting the sum rule could be relevant. The y then implemente d thi s idea [248] , not b y provin g the su m rul e i n this generalit y bu t usin g the su m rul e fo r nic e V' s and takin g suitable limits . At th e sam e time , I wa s encouragin g worker s i n th e field t o find example s o f mixed spectrum i n this 1 > a > \ region . I t wa s known one could have point spec - trum mixe d i n the a.c . spectrum [795 , 973], but th e issu e was singular continuou s spectrum an d what kind . I n honor o f the change o f millennium, I produced a list o f open problems in 199 9 and high in the list was mixed singular continuous spectru m with powe r decay . In January 2001,1 learned o f a paper by a young Russian, Serge y Denisov [271] . In it , h e showe d ther e existe d L 2 potential s V wit h essentiall y arbitrar y singula r part o n [0 , EQ] fo r EQ < oo. Technically , this didn't solv e the problem I stated sinc e I aske d fo r a power bound (thi s was later don e b y Kiselev [638]) , but I regarded i t as essentially a solution. Nic k Makarov an d I were so impressed b y this result tha t we arranged a postdoc positio n fo r Denisov , o f which mor e later . That summer , Rowa n Killi p visite d Caltec h an d w e tried t o understan d wha t made Denisov' s proo f work . W e weren't abl e to understand th e technical details o f his proo f becaus e h e use d Krei n systems , whic h w e had neve r see n an d fo r whic h the existing literature wa s spotty. Bu t w e understood that h e was using some kin d of invers e theor y construction . Initially , thi s puzzle d me . Fo r years , I' d bee n tol d that on e shoul d us e invers e spectra l theor y t o construc t mixe d spectra , an d m y response wa s to poin t ou t tha t th e Marchenk o theor y wouldn' t wor k i n such case s and tha t th e Gel'fand-Levita n theor y provide d n o informatio n abou t asymptotic s at infinity . Nevertheless, Deniso v prove d hi s potentia l wa s i n L 2. A t first, w e though t the magi c mus t b e i n Krei n systems , bu t the n w e realize d h e wa s gettin g th e L 2 PREFACE T O PAR T 1 Xlll property fro m a su m rule . S o w e focuse d o n usin g su m rule s goin g beyon d thei r use in Deift-Killip. I n connection with the earlie r wor k and it s followup, Killi p ha d learned o f the su m rule s o f Cas e [190 ] an d variou s revie w article s o f Nevai [814] . With these as background, w e were able to handle discrete eigenvalues in the Jacob i case an d prov e a general su m rul e that describe d exactl y whic h spectra l measure s 2 corresponded to Jacobi matrices with ^2 n{an — l) + 6 ^ < o o [633] . Alon g the way, I learne d tha t du e to wor k o f Szeg o an d Shohat , thi s proble m ha d bee n solve d i n the earl y part o f the twentiet h centur y unde r th e additiona l stron g conditio n tha t supp(d/i) C [—2,2] . Bu t I put asid e understanding ho w they ha d don e s o while w e worked ou t som e o f the detail s o f our work . In the fal l o f 2001, Denisov showed up at Caltec h and, with my encouragement , gave a course on Krein systems. Sinc e OPUC ar e a discrete analo g that motivate d Krein, Deniso v spen t th e firs t fe w week s presenting thei r theory , followin g i n par t the chapte r i n Akhiezer' s boo k [17 ] ( I had, o f course , rea d thi s boo k whe n I wa s writing m y articl e o n th e momen t proble m fo r R bu t I ha d skippe d thi s chapte r as irrelevan t t o me!) . I wa s struck b y the beaut y o f the subjec t an d bega n t o tr y to understan d i t better . I quickl y foun d th e proo f I giv e o f the Szeg o recursion , wondering wh y Akhiezer' s proo f wa s s o complex. I eventuall y foun d thi s proo f i n Atkinson [60 ] an d Landa u [672] , bu t i t wa s unknow n enoug h tha t I wa s abl e t o surprise on e expert wit h it . At the same time, in connection with my work with Killip, I realized what we' d found wa s an analog o f Szego's theorem, which I began to study. I also realized tha t OPUC wa s a n analo g o f Schrodinge r operators , a virtua l playgroun d fo r spectra l theorists. B y February o f 2002, I was convinced that th e sensibl e way to carry ove r all the Schrodinge r operator s t o OPU C wa s i n a singl e article , rathe r tha n lot s o f short ones . I als o realize d thoug h tha t OPU C ha d n o summary sinc e Geronimus ' book forty year s before, an d eve n that ha d focuse d o n only part o f the then know n subject. Th e ide a bega n t o for m o f writin g tw o revie w article s o n OPUC , eac h I estimated to be 120-15 0 pages long, the firs t reviewin g the classica l theory an d th e second o n spectral theory . As I began planning the articles, I began to wonder who had first writte n abou t Szego's theorem a s a sum rule. I asked Paul Nevai , who replied that h e didn't kno w — but i t wa s a n interestin g questio n an d s o he sen t ou t a n e-mai l blas t t o abou t a doze n experts . On e fro m th e Russia n traditio n replie d tha t h e wasn' t sur e an d that h e hadn't see n the paper , bu t he' d hear d i t migh t b e " ..." an d h e gav e th e reference t o Verblunsk y [1067] . I got hol d o f this an d th e earlie r [1066 ] an d rea d them with fascination . I t wa s hard going , but a s I absorbed th e papers, i t becam e clear that ther e was an enormous number o f ideas in these papers that ha d becom e important, bu t the n forgotte n an d late r rediscovered ! S o added to the agend a wa s making sur e Verblunsky go t the credi t s o long denied him ! I began a n e-mai l correspondenc e wit h Golinski i an d Neva i about thei r paper , and i n discussin g th e GG T representation , Golinski i tol d m e abou t th e pape r o f Cantero, Moral , an d Velazque z [181 ] written i n 200 0 but onl y publishe d i n 2003 . He told m e the basi c ide a that orthonormalizin g {l,z,z _1,2:2,2:_2,...} lead s to a basis expressible in terms o f the usual OPUC and a five-diagonal matrix expressibl e in term s o f th e Verblunsk y coefficients . Withi n a fe w hours , I ha d rederive d th e formulae an d then ove r a period o f a fe w weeks, Golinskii and I , via e-mail, worke d out th e detail s fo r genera l spectra l theor y tha t appea r a s Sectio n 4.3 . M y projec t xiv PREFAC E T O PAR T 1 had becom e large r a s I realized that presentin g the CM V matrix an d exploitin g i t was also a part o f the mix . As the summer o f 2002 finished, I realized I still had a lot to do and the idea of two 150-page articles wouldn't suffice . A book was needed, and when Sergei Gelfan d offered t o publis h i t i n th e sam e serie s wher e Szego' s boo k [1026 ] appeared , th e temptation t o accep t withou t consultin g other publisher s wa s s o great. Slowly the project too k on extra elements that caused it to grow. I had include d background section s on Schu r functions an d o n OPRL to make the book accessibl e to spectral theorists, but Doron Lubinsky convinced me to add a background section on operator theor y to make it accessibl e to workers with a background i n OP. Th e desire to reac h both group s produce d backgroun d materia l i n man y place s that, I hope, als o makes it accessibl e to graduate students . Other factor s cause d manuscrip t growth . I realized that ther e were results lik e the Bello-Lope z extensio n o f Rakhmanov' s theore m tha t I neede d t o includ e i f I wanted th e boo k t o b e a s comprehensiv e a s I desired . Perc y Deif t kep t pushin g me to include al l the main proof s o f the strong Szeg o theorem. Onc e the book ha d grown a s i t had , I foun d i t appropriat e t o ad d a chapte r o n use s o f OPUC i n th e theory o f OPRL . In this way , two 150-pag e articles gre w to on e book gre w to tw o volumes. The table o f contents and list o f notation cover both volumes. T o save trees, the author index , subjec t index , an d th e bibliograph y i n this volum e ar e onl y fo r thi s volume, althoug h ther e i s a complete bibliograph y an d complet e indice s i n Part 2 . For consistent numberin g betwee n volumes, the bibliograph y i n Part 1 has gaps i n its numbering . I considere d mor e elegan t referenc e referra l wit h autho r an d yea r but decide d agains t it . Whe n ther e ar e fou r o f fiv e "S " references , i t i s eas y t o locate [ST97] , but whe n ther e ar e ove r a hundred "S " references, i t i s difficult. I n the end, functionalit y wo n ove r elegance . I warn the reade r o f a personal quirk . I' m tol d that prope r usag e requires th e addition o f a period in a sentence that ends with a set-out equation. Bu t I find extr a dots i n suc h equation s confusing , s o I don' t us e punctuatio n i n set-ou t formulae , even i f proper gramma r say s they shoul d b e there. I doub t tha t thes e book s wil l hav e th e fou r edition s tha t Szeg o [1026 ] did , but i t seem s likel y ther e wil l b e late r editions . Tha t mean s I especiall y wel - come comments , corrections , missin g topic s an d references , an d informatio n o n new papers . Fo r th e latter , I muc h prefe r a lin k t o a n onlin e archiv e rathe r tha n that yo u sen d m e attachments . Addend a an d majo r correction s will be poste d a t http: //www. math. caltech. edu/opuc. html. M y email i s bsimonOcaltech. edu.

One perso n wh o learne d abou t thi s boo k wondere d ho w I coul d eve n Te X s o many page s in "onl y two years." Th e secret i s that I didn't. I wrote out lon g han d and Cheri e Galvez , m y super b secretary , TeXe d th e manuscript . I a m indebte d for he r har d wor k an d competence . Rowa n Killi p provide d advic e t o he r o n Te X technical issues , and I'm gratefu l fo r hi s help on this. Many mathematician s provide d usefu l input , bu t I should begi n with fou r de - serving specia l mention . Rowa n Killi p provide d insigh t o n man y o f th e issue s I studied here . Perc y Deif t wa s merciles s i n pushin g m e t o includ e materia l tha t worked out wel l and has added to the usefulness o f the book. Frit z Gesztesy taugh t me muc h abou t periodi c Jacob i matrices . Mos t o f all , Leoni d Golinski i spen t a PREFACE T O PAR T 1 x v huge amount o f time goin g through the manuscript. H e not onl y foun d typos , bu t places where the argument s weren' t quit e right . I also want to thank Richard Askey , Alexei Borodin, Albrech t Bottcher , Danie l Bump, Alicia Cachafeiro, Danny Calegari, Tiberiu Constantinescu, David Damanik, Sergey Denisov, Nathan Dunfleld, Harr y Dym, Jef f Geronimo , Dimitri Gioev, Peter Harremoes, Mourad Ismail, Alain Joye, Thomas Kailath, Victor Katsnelson, Serge y Khrushchev, Pete r Kuchment , Ar i Laptev , Yora m Last , Danie l Lenz , Guillerm o Lopez Lagomasino , Doro n Lubinsky , Russel l Lyons , Alphons e Magnus , Francisc o Marcellan, Andre i Martinez-Finkelshtein , Pete r Miller , Irin a Nenciu , Pau l Nevai , Olav Njastad , Fran z Peherstorfer , Yur i Safarov , Miha i Stoiciu , Gunte r Stolz , Alexander Teplyaev , Joh n Toland , Vilmo s Totik , Doro n Zeilberger , an d Andre j Zlatos fo r usefu l input . Finally, m y wif e Martha, whos e lov e makes it al l easier .

Barry Simo n Los Angeles, August 200 4 This page intentionally left blank Notation

This lis t o f notation i s broken int o the Gree k an d Roma n alphabet s an d the n nonalphabetic. Sinc e ther e ar e onl y a finite numbe r o f letters, som e symbol s ar e used i n differen t ways !

Greek Alphabe t i n the followin g order : a, /?, 7, T, > ^> ^

an Verblunsk y coefficients ; se e (1.1.8 ) Pf solutio n o f 0 + /T1 = £* wit h |/3 | > 1 ; see (13.8.21 ) (3(zo) Jitomirskaya-Las t ratio ; se e (10.10.13 ) d(32n,k(@) measur e i n varying ratio asymptotics ; se e (9.8.7 ) 7(2) Lyapuno v exponent , 7(2: ) = lim n_+oo \ lo g ||Tn(z)||; se e (10.5.14 ) ra?A essentia l spectru m fo r singl e arc , T a,\ — {z G <9 B | |arg(A^) | > 2arcsin(a)}; se e (9.9.2 ) re(x) functio n o f Damanik-Killip , r e(a;) = §[u(# ) — v(x)]; se e th e Note s t o Section 13. 1 m T0(x) functio n o f Damanik-Killip , V 0(x) = —^[u(x) + v(x)] , see th e Note s t o Section 13. 1 r(Bj) contou r surroundin g BJ; see (11.7.16 ) T(Gj) contou r surroundin g Gj\ se e (11.7.16 ) T{z) analyti c functio n whos e rea l par t i s th e Lyapuno v exponent , 7(2) ; se e (10.11.18) 50D relativ e Szeg o function; se e (2.9.6 ) A(z) Freu d function ; se e Theorem 2.2.1 4 A (2) discriminan t fo r periodi c Verblunsky coefficients ; se e (11.1.2 ) Qn poin t mas s approximations ; se e Theorem 2.2.1 2 dr)n(z) varyin g measure; se e (9.8.1 ) 9n(zo) Priife r angl e fo r OPUC ; se e (10.12.1 ) Qj CM V building block , 9, = (% _ % ); se e (4.2.20 ) n 1 nn leadin g coefficien t o f (p n(z) = n nz -f - • • •, K n = H^H"" ; se e (1.5.1 ) ^oo(C) Christoffe l function , Aoo(£ ) = inf{/|7r(e z6>)|2 d//(0) | 7 r a polynomial , TT(C) - 1} ; see (2.2.2 ) Xj(A) eigenvalue s o f a compact operato r A; see (1.4.27 ) 26> 2 ^n(C) approximat e Christoffe l function , X n(C) — min{/|7r(e )| d/x(0) | deg7r < n, TT(C ) = 1} ; see (2.2.1 ) 20 p An(£, d//;p) ^Christoffe l function , \ n((,d/j,',p) — min{/|7r(e )| d/x | deg7 r < n , TT(C) = 1} ; see (2.5.1 ) f^x\z) Schu r functio n o f an Aleksandro v family ; se e Subsection 1.3. 9 xviii NOTATIO N

F(x\z) Caratheodor y functio n o f an Aleksandrov family ; see (1.3.90 ) <3>^ OP s o f an Aleksandro v family ; se e (3.2.1 ) dp\ measure s o f an Aleksandro v family ; se e (1.3.91 ) A unitar y perturbation , A : PH - > PH, V = UAP + U(l - P) ; se e (4.5.10 ) pn(A) singula r value s o f an operator A, i.e., eigenvalue s o f \A\; see (1.4.33 )

d/iac absolutel y continuous part o f the measure d\i, i.e. , i f dp, = w^ - f d/xs, then

dpac = w^; se e Subsection 1.4. 6 see d/ipp pur e poin t par t o f a measure, dp pp = J2 X Mi^})^; Subsectio n 1.4. 6 dps(9) singula r par t o f a measure, dp s — dp — dpac; see Subsection 1.4. 6 dpsc singula r continuou s par t o f a measure, dp sc = dp s — dp pp; se e Subsec - tion 1.4. 6 dpfftf) G I approximations t o a measure; se e (6.1.24 ) v{n) Beurlin g weight ; se e (5.1.38 ) dv densit y o f zeros measure, dv = limdv n i f it exists ; se e Section 8. 1 dvn finite densit y o f zeros, dvn = ^ Yll=i ^ wher e Zj ar e the zero s o f $n(z) counting multiplicity ; se e Example 1.7.1 7

dvnj{6) approximation s use d i n Khrushchev's theory , dv n^(6) = . ( eie\\2 |f; see (9.3.14 )

En(z) vecto r recursio n fo r OPUC ; se e (2.2.44 ) 7r* extende d abelia n periods ; se e (11.8.2 ) 7rn decayin g CM V function, 7r n = Tn + F{z)xn\ se e (4.4.5 ) Hk(Bj) k-th periodi c associate d t o band, BJ; se e (11.7.16 ) Uk(Gj) fc-th periodi c associate d t o gap, Gj\ se e (11.7.16 ) 2 1 2 Pi p j = (l-\aj\ ) / ; se e (1.5.21 ) p(B) resolven t se t o f an operator B, p(B) = {z \ (B — z) 1 exists}; se e Subsec- tion 1.4. 2 p(Tn) Kotani-Ushiroy a ratio ; see (10.5.83 ) (Td(A) discret e spectru m o f an operator A; se e Subsection 1.4. 6 cress(^) essentia l spectru m o f a n operato r A, aess(A) = cr(A)\ad(A)] se e Subsec - tion 1.4. 6 n{ se e an hal f o f CM V basis, an = \2n — z~ P\ni (4.2.8) cr(B) spectru m o f an operator B, ^ lef t polynomial s i n the theory o f matrix-valued measures ; se e (2.13.17 ) n $* (2) dua l o f monic OPUC, z $n(l/z); se e (1.5.9 ) Xa{g) grou p character ; se e Section 6. 3 Xn CM V basi s element; se e (4.2.5 ) Xn fre e CM V basis element; see (4.2.4 ) NOTATION xi x

#n secon d kin d OPUC ; se e (3.2.3 ) Q± wav e operators; se e (10.7.46 ) an d (10.7.80 )

Roman Alphabe t an Jacob i parameter ; se e (1.2.13) an(x) continue d fractio n coefficients ; se e (11.9.1 ) ab(z) normalize d coefficien t i n quadratic equatio n fo r F\ se e (11.7.66) A suppor t o f the a.c. part; se e (10.10.4) A se t where j(E) = 0 for subshifts; se e Section 12.8 An Wal l polynomial ; se e (1.3.65) Avn(f) ergodi c average ; se e Theorem 10.5A. 2 A alphabe t fo r subshift; se e Appendix 10.5 l AR annulus , {z | R~ < \z\ < i?}; see (7.1.1) 21 Abe l map ; se e (11.7.26 ) 21 x extende d Abe l map ; se e (11.8.5) 2l„ two-side d Beurlin g algebra , {a | J^n(l + |n|)^|a n| < oo} ; see (5.1.44) 21+ positiv e Beurlin g algebra , {a G 21^ | a(n) = 0 for n < 0}; see (5.1.44) 2l~ negativ e Beurlin g algebra , {a £ 21 ^ | a(n) = 0 for n > 0}; see (5.1.44) bn Jacob i parameter ; se e (1.2.13) bni£ O P ratio error ; se e (9.1.1) 6n(z, d/i) invers e Schu r iterates ; se e (9.2.14) bb(z) normalize d coefficien t i n quadratic equatio n fo r F; see (11.7.67) B se t of z's in <9IE D for which lim r|i F(rz) doe s not exist; se e (10.10.5) B se t fo r whic h limsup:r n(2:) < o o in theory o f subshifts; se e (12.8.3 ) Bj band s fo r periodi c Verblunsk y coefficients , se e (11.1.6 ) Bj unio n o f touching bands; se e Section 11. 7 Bn Wal l polynomials; see (1.3.66 ) B(f) rati o D/D; se e (6.2.40 ) B(fji,v) bilinea r potential ; se e (8.1.9) B facto r i n CMV proof o f Geronimus relations ; se e (13.2.16 ) %nQ cn moment s o f a measure, J e~ d\i{&)\ se e (1.1.20 ) cn O P half o f solution o f the Geronimo-Case equations ; se e (13.6.10 ) cn>£ O P ratio; se e (9.1.13 ) cb(z) normalize d coefficien t i n quadratic equatio n fo r F; se e (11.7.68) cap(if) logarithmi c capacit y o f a compact se t K\ se e Section 8. 1 C unio n o f singula r continuou s support s ove r a n Aleksandro v family ; see (10.10.3) Cn normalize d solutio n o f Geronimo-Case equations ; se e (13.6.42 ) Cf0 th e contour s \z\ = 1 on the Rieman n surfac e convolution; see (6.2.10) C(z,w) complet e Cauch y kernel , '^zr z\ se e (1.3.15 ) C th e complex number s C+ {zeC\Imz>0} xx NOTATIO N

Cij(dn) CM V matrix; se e (4.2.12 ) Cij(dfx) alternat e CM V matrix; se e (4.2.13 ) dn(A) Hausdorf f dimensio n o f a set A; se e (2.12.8 ) djti Taylo r coefficient s o f D(z)\ se e (7.2.4 ) dj^-i Taylo r coefficient s o f D(z)" 1; se e (7.1.5 ) dn O P squar e root integral ; se e (9.1.14 ) deg(/) degre e o f meromorphic function , i.e. , the number o f solutions o f deg(/) = w fo r generi c w G C; se e (11.7.11 ) det Predhol m determinan t o f a trace clas s operator; se e (1.4.64 ) det2 renormahze d Predhol m determinan t o f a Hilbert-Schmid t operator ; se e (1.4.82) D&c{.e%e) th e function i n L 2(<9D, dp), whic h is the boundary valu e o f D(z)~ 1, se t t o zero on a support fo r d/x s; se e (2.4.33 ) Dn{dfi) Toeplit z determinant ; se e (1.3.12 ) Dn(f) Dirichle t approximation ; se e (2.12.32 ) DA Dira c operator; se e the Note s to Sectio n 13. 1 D(z) Szeg o function, exp( / f£±f log(w(6))§ ); se e (2.4.2 ) D^(ZQ) loca l infinitesimal Hausdorf f dimension ; se e (10.8.28 ) ED uni t disk , {z | \z\ < 1 } BR {z | \z\ < R} D°°'c se t o f Schu r parameters; se e Subsection 1.3. 6 T>({Gj}) Dirichle t dat a torus ; se e Section 11. 4

Ef eigenvalue s o f Jacobi matrices; se e (13.8.21 )

En Coulom b energy ; se e (6.4.2 ) E probabilit y expectatio n £ extende d CM V matrix; se e (10.5.34 ) £o(J) energ y term i n C o sum rule, £o(J) = ]Cj, ± l°&\Pf(J)\i se e (13.8.52 ) £q(0) periodize d CM V matrix; se e (11.2.4 ) £(dfjb) Coulom b energy fo r potential theory £(dfjb) = J log\z — wl"1 dji(z) d/j,(w); see (8.1.7 ) f+(z) use d fo r Schu r functio n whe n two-side d sequence s ar e involved ; se e (10.11.24) /_(z) lef t sid e Schu r function ; se e (10.11.24 ) an d Theore m 10.11.1 6 fa(z) Schu r functio n fo r Geronimu s polynomials , f a(z) = ~2az~ ° — » see (1.6.82 ) an d (9.5.2 ) fn(z) Schu r iterates ; se e (1.3.37 ) fin\z) Schu r approximants ; se e (1.3.41 ) f(k) Jos t functio n fo r Schrodinge r operators ; se e (10.7.29 ) f(x,k) Jos t solutio n fo r Schrodinge r operators ; se e (10.7.26 ) f(z) Schu r functio n fo r a measure 1 ^^W = / f^r f dfj,(6); see Section 1. 3 2 2 F functio n i n P 2 su m rule ; fo r E > 2 , F(E) - \ $^\E - 4\V dE\ se e (13.8.27) an d (13.8.33 ) Fn(f) Feje r approximation ; se e (2.12.33 ) F^) coefficient-strippe d Caratheodor y function ; se e (3.4.14 ) an d (3.4.18 ) F(z) Caratheodor y function o f a measure, F(z) = J firz ^ dp{6)] see Section 1. 3 NOTATION xx i

1 71 F(d/jJ) leadin g term i n limit o f Toeplitz matrices , F(djj) = limn_oo D^d/i) / = njlot1 - Kf); see Theorem 2.1.2 ie e T Fourier-CM V transform , {Tj) n = J Xn(e ) f(e\ ) d//(0) ; see (10.7.79) JF0 fre e Fourier-CM V transform , T fo r d\i = ff; se e (10.7.79) Tkt{dfj) ful l GGT matrix; se e (4.1.25) and Proposition 4.1.3 g matri x Schu r function ; se e (4.5.13) go unperturbe d matri x Schu r function ; se e (4.5.14) gn approximat e Jos t functio n hal f o f a solution o f Geronimo-Case equations ; see (13.6.13 ) gn(6,dfi) functio n neede d i n Khrushchev theory ; se e (9.2.32) G auxiliar y functio n fo r P2 sum rule give n b y G{a) = a 2 — 1 — log (a2); see (13.8.28) Goo limi t valu e fo r Geronimo-Case equations ; se e (13.6.43) GQ{Z) unperturbe d matri x Caratheodor y function ; se e (4.5.12) 2 Ga,\(z) allowe d limi t fo r ratio asymptotics , G a,\(z) — |[(1 -f- \z) + [( 1 - \z) + 4a2Az]1/2]; se e (9.5.5) Gj gap s in periodic essentia l spectrum ; se e Section 11.7 Gn normalize d hal f solutio n o f Geronimo-Case equation ; se e (13.6.42) Gn(z) functio n studie d b y Golinskii, G n(z) = F(z)$ n(z) + ^n(^); se e (3.2.40). Up to a constant, thi s i s the Uk of (9.2.28) . Gn{z) forma l dua l functio n t o Golinskii' s G n, G n(z) = F(z)$ n — $n(z); se e (3.2.41). U p to a constant, thi s i s the u*k of (9.2.28) . G{dji) secon d ter m i n Toeplit z determinan t asymptotics , G{djj) = njlo( l ~ ~ l^il2)"7""1; se e (2-L3) G(z) matri x Caratheodor y function , G(z) = P[^f ]P ; se e (4.5.11) G(z) /^Sp ; se e (10.1.5) G(z) Green' s functio n fo r £; see (10.11.25) Goo noneigenvalu e set ; see Theorem 10.1. 5 Gki({an}%Lo) GG T matrix; se e (4.1.4) and (4.1.5) Gn MN T integral operator ; se e (9.4.28) ha(A) Hausdorf f a-dimensiona l measure ; se e (2.12.8) hjsf Hanke l determinant ; se e (1.2.29) h(a) Hanke l operator ; se e (6.2.2) h(a) Hanke l alternat e operator ; se e (6.2.2) x 2 1 2 2 2 H ' Sobole v space , H / = { / 6 L (<9D, f£) | £|fc||/fc| < 00}; see (6.2.36) H2, H°° Hard y space s H(f) Hanke l operato r wit h symbo l /; se e (6.2.6) 2 H&c L (<9D, d/iac); se e Subsection 1.4.7 Hpp spac e o f eigenvectors; se e Subsection 1.4.7 2 Hsc L (<9D,dfisc)i se e Subsection 1.4. 7

/(/) log(D/D); se e (6.2.39) Ji trac e class , {^ 4 | Tr(|^4|) < 00} ; see Subsection 1.4.1 2 I2 Hilbert-Schmid t ideal , {A | Tr(A*A) < 00} ; see Subsection 1.4.1 3 2^ invarian t measures on <9 B for a measure /i on U(l, 1); see Proposition 10.6. 2 p lp trac e ideal , {^ 4 | Tr(|A| ) < 00} ; see Subsection 1.4.1 3 xxii NOTATIO N

1(A) se t of measures o n <9 D left invarian t b y the projection actio n o f A\ se e Theorem 10.4.1 2 J J = (J -i )> relevant to the definition o f U(l, 1); see (10.4.1) J^ onc e strippe d Jacob i matrix ; se e Subsection 1.2.1 2 JM Jacob i matrix ; se e (1.2.17) Jn normalizatio n facto r o f 3>^ in the theory o f matrix-valued measures ; see (2.13.22) Jn\F n x n matri x obtaine d fro m uppe r righ t corne r o f a Jacob i matrix ; se e (1.2.61) J(S) Jacob i variety o f the Rieman n surfac e 5; see (11.7.25) 1 Je restrictio n o f S(C + C~ )S to the even subspace; se e Theorem 13.2. 1 Jo restrictio n o f S(C + C_1)5 t o the odd subspace; se e Theorem 13.2. 1 J+ restrictio n o f J(C + C~1)J~1 t o the even subspace; se e Theorem 13.2. 2 J~ restrictio n o f J(C + C~l)J~l t o the odd subspace; see Theorem 13.2. 2 JX(S) extende d Jacob i variety; se e (11.8.5)

Kn normalizatio n facto r o f )\ ', se e (6.1.16) M^j matri x invers e to matrix-valued Toeplit z matrix ; see (2.13.14) M(ao,..., ctp-i) modulu s o f a point i n Dp; see (11.4.29) Mf reflectio n ma p associated to CMV, M, (Mf)(z) = f(z); se e (13.2.1) M(z) M-functio n fo r analyzing effect o f stripping on meromorphic Caratheodor y

functions, M(z) = z(l + a0)(l + F(z)) + ( 1 + a0)(l - F(z))\ se e (11.7.76) wM+,i(X) probabilit y measure s o n a compact Hausdorf f space , X Mij(dfi) M hal f o f CMV CM factorization , Mij = (xi, Xj)'i see (4.2.16 ) Mq((3) periodize d M matrix ; se e (11.2.7)

16 1 2 l0 t ie NA^ ) nor m associate d to a projective actio n o f A, if UQ = 2~ / (e ) 1 N a(e ) = \\Aue\\i se e (10.4.30) 0(f; ZQ) degre e o f a meromorphic function ; se e (11.7.11) n se e 0(UJQ) orbi t o f a dynamical system , O(u> 0) = {^ ^}^L-ooJ (10.5A.1 ) pn normalize d OPRL ; se e (1.2.5) NOTATION xxii i pn decayin g CM V function , p n = y n + F(z)x n; se e (4.4.4 ) P pur e poin t support , U x{z | /XA({^} ) ^ 0 } fo r a n Aleksandro v family ; se e (10.10.1) P+ fundamenta l projectio n i n Wiener-Hopf theory ; se e Section 5. 1 P_ 1 — P+ i n Wiener-Hopf theory ; se e (5.1.8 ) Pn moni c OPRL ; se e (1.2.5 )

Pr(0,

V Poisso n ma p i n MNT theory , V(F, z) = J^ |^], 2F(e^)^; se e (9.4.29 ) iG P(di/) dua l o f Poisso n map , M+ tl(B) - > M+ ti(dD) b y / f(e )V(dv)(0) = ie i(p i(p f P r(e \e )f(0)dv(re )i se e Proposition 8.2. 2

qn(x) secon d kin d OPRL ; se e Subsection 1.2.1 0 2 n (n)q ^factorial , (n) q = ( 1 - q)(l - q )... ( 1 - q ); se e (1.6.42 ) 7 ( [ j ) q ^-binomia l coefficient , [ j]q = (i)g (^V)g; se e (1.6.43 ) Qn,m(%,y) polynomia l use d i n Lopez theory ; se e (9.9.10 )

Qn(x) secon d kin d OPRL ; se e Subsection 1.2.1 0 Q(J) entrop y i n P2 sum rule ; see (13.8.34 ) Q rationa l number s

rn(x) remainde r i n continued fractio n expansio n o f a real number x ; see (11.9.1 ) rn(z) OPU C normalize d Priife r radius ; se e (10.12.3 ) Rn(z) OPU C Priife r radius ; se e (10.12.2 ) RD R D — radius o f convergence o f D i f dfi s = 0 ; see (7.1.1 ) 1 x jRp-i RD- — radius o f convergence o f D~ i f d/i s — 0 ; see (7.1.2 ) RF Rp — radius o f convergence o f F(z) abou t z = 0 ; see (7.1.1 ) PNT Nevai-Toti k radius ; se e (7.1.3 ) R&* R$* = sup{ r I sup njz|

n{ sn hal f o f alternate CM V basis, s n = a?2 n = z~ Pin\ se e (4.2.10 ) sn(f) w-t h Taylor coefficien t o f /; se e (1.3.42 ) s(fjt) entrop y o f // i n Furstenberg's theorem ; see (10.6.10 ) S combine d singula r support s o f an Aleksandro v family ; se e (10.10.2 ) Sais(A) Hausdorf f measur e constructor ; se e (2.12.7 ) S\ suppor t o f singular par t o f dfj,\] see (10.1.2 ) S(a) matri x use d i n th e Killip-Nenci u proo f o f th e Geronimu s relations ; se e (13.2.13) S(/JL I v) relativ e entropy; se e (2.3.2 ) S(z) S(z) = Zn=o anz n; se e (7.2.3 ) Sz(£) Szeg o transform o f measures; se e (13.1.4 ) SL(2, C) 2x 2 comple x matrices o f determinant 1 xxiv NOTATIO N

SL(2, R ) 2x 2 rea l matrices o f determinant 1 SU(1,1) Ae SL(2 , C) s o that A * J A = J; se e (10.4.1 ) SU(l,l;Jr) A G SL(2,C ) s o tha t A*J rA = J r; i n fact , i t equal s SL(2,R) ; se e Proposition 10.4. 1 S Schu r map, from Schu r functions to Schur parameters; see Subsection 1.3. 6 S Rieman n surface ; se e Section 11. 7 S ma p used in Killip-Nenciu proof of the Geronimus connection; see (13.2.14) (4.2.9 ) t(a) Toeplit z operator ; se e (6.2.2 ) lm Too(z) limi t o f modified transfe r matrice s with a n G £ , see (10.7.7 ) T*>f Toeplit z matrix ; se e (1.3.11 )

Tn{z) transfe r matrix ; se e (3.2.27 )

Tn{z) modifie d transfe r matrix ; se e (10.7.3 ) ^ti,...,^^) multicharacte r fo r SU(n); se e (6.3.6 ) T(f) Toeplit z operato r wit h symbo l /, se e (6.2.6 ) Tr(A) trac e o n trace class ; se e Subsection 1.4.1 2 T ma p used in Killip-Nenciu proof o f the connection formulae o f Berriochoa, Cachafeiro, an d Garcia-Amor ; se e (13.2.36 ) Uk to p half o f decaying solutio n o f transfer matrix , Uk = ifik + ^(^) (Pfc5 see (9.2.28) ul botto m hal f o f decaying solutio n o f transfer matrix , ul = —^ + F(z)(p^; see (9.2.28 ) un rati o use d i n invers e Geronimu s relations , u n = ^n+2/^n+i 5 se e (13.1.34 ) and (13.1.37 ) un(z, J) Jos t functio n fo r OPRL ; se e Section 13. 6 U^(x) potentia l o f a measure / i on C ; see (8.1.8 )

U(P, Q) ma p i n Kato similarity transforms , U(P 1 Q) = J2j=i QjPj'i se e (12.1.10 ) U(l, 1 ) A G GL(2, bbC) so A * = J A = J; see Section 10. 4 U(n) n x n unitar y matrice s see vn rati o used in inverse Geronimus relations, v n = —^n+2/Vn+i 5 (13.1.34 ) and (13.1.37 ) Vk canonica l basis of first kind holomorphic differential s o n the Riemann sur - face «S ; se e Proposition 11.7. 4 y(n) Verblunsk y remainde r term ; se e (1.5.53 ) V(P, Q) ma p i n Kato similarit y transform , V(P , Q) = ]Cj=i PjQj'i see (12.1.11 )

V(6) densit y o f zero s fo r periodi c problems , V{6) — - (4 JA2rigxl1/2; se e (11.1.19) Wn Wey l solution; se e (13.9.5 ) w{9) ?ffi, i.e. , dfi = w(0)& + d/i s; see (1.1.5 ) Wi vecto r componen t o f CMV solution ; se e (4.4.21) W(ip) Coulom b interactio n o n <9D ; see (6.4.4) W(6) OPU C boos t group ; se e (10.14.5 ) NOTATION xx v xn alternat e CM V basis; see (4.2.6) free alternat e CM V basis; see (4.2.6 ) Xp Floque t space ; see (11.2.2 ) yn secon d kin d alternat e CM V basis; se e (4.4.2 )

Za s a = ±±f ; see (11.1.30 ) se e Zjn j-th zer o o f $ n{z)\ Exampl e 1.7.1 7 ^oo (djj) limi t points o f zeros o f $n an d zeros o f some $n; see the remark followin g Theorem 1.7.1 2 Zn(&\±) zero s o f n; see the remark followin g Theore m 1.7.1 2 Z\Xd\i) limi t point s o f zeros o f 3> n; see the remark followin g Theore m 1.7.1 2 ^SL(^) strictl y limit s o f zeros o f $n; se e the remark followin g Theore m 1.7.1 2 Zu zer o set fo r 7; see (10.11.1 ) Z{J) entrop y i n C o sum rule; se e (13.8.22 ) Z(J I J^) step-by-ste p entrop y i n C o sum rule; see (13.8.24) Z integer s {0 , ±1, ±2,... } Z+ nonnegativ e integer s {0,1 , 2,... } Nonalphabetic \A\ absolut e valu e o f an operator; se e (1.4.28 ) (*; o)oo — rijlo(l"~ w Qj) use( l f° r theta function ; se e (1.6.54) (S) grou p semidirec t produc t { •, • } Poisso n bracke t o n Dp; see (11.11.9 ) {•, • } o fre e Poisso n bracke t o n B; see (11.11.1 ) n Qn(z) reverse d polynomials , Q*(z ) = z Q n(\jz)\ se e (1.1.6) f "restriction " / \ A i s the function (o r operator), /, restricte d to a set A A se t differenc e AAB = (A\B) U (B\A) This page intentionally left blank This page intentionally left blank Bibliography

Note on Russian Names: Anyon e writin g abou t a subjec t lik e OPU C o n whic h there have been substantial Russian (an d Ukrainian) contributions has to deal with the issu e illustrated b y the fac t tha t i n their Englis h translations , th e name i n th e book fo r th e autho r o f [17 ] i s Akhiezer , whil e th e nam e o f th e autho r i n [13 ] i s Achieser! I hav e decide d t o us e i n bot h reference s an d object s (e.g. , Chebyshe v polynomials) a consistent spelling . But the reader i s warned that thi s may produc e a difficult y i f you ar e trying t o orde r a boo k o n Amazo n usin g a n autho r spellin g from thi s bibliography .

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Bach, V. , 299 , 42 7 Cachafeiro, A. , 8 , 134 , 171 , 426, 42 8 Badkov, V. , 89 , 90, 151 , 427 Calvetti, D. , 413, 42 6 Baik, J. , 7 , 42 7 Cantero, M. , xiii , 135 , 239, 262 , 273, 274, Bakonyi, M. , 61, 69, 216 , 262 , 42 7 430 Barbey, K. , 156 , 42 7 Caratheodory, C. , 37 , 38, 43 0 Barrios, D., 286 , 388, 389, 42 7 Carey, R. , 333 , 430 Bart, H. , 333 , 427 Carleman, T. , 55 , 430 Bartoszynski, R. , 410 , 42 7 Case, K.M. , xiii , 10 , 177 , 344, 430, 43 5 Basor, E. , 90 , 332, 333, 344, 42 8 Charris, J. , 178 , 43 0 Baxley, J. , 136 , 42 8 Chebyshev, R, 24 , 43 0 Baxter, G. , 11 , 116, 313, 317, 331, 332, 42 8 Chen, Y. , 90 , 136 , 333, 42 8 Bello, M. , 8 , 98 , 42 8 Chihara, T. , 24 , 43 1 Belokolos, E. , 216 , 42 8 Chow, Y.S., 410, 43 1 Benderskii, M. , 409 , 42 8 Chowdhury, D. , 352 , 43 1 Berezanskii, J. , 286 , 42 8 Christ, M. , xii , 43 1 Berezin, F. , 136 , 42 8 Christoffel, E. , 24 , 43 1 Berg, C. , 136 , 42 8 Clark, S. , 216 , 43 1 Berg, L. , 333 , 428 Cohn, A. , 107 , 43 1 Bernstein, S. , 88 , 134 , 42 8 Combes, J.M. , 299 , 43 1 458 AUTHOR INDE X

Constantinescu, T. , 61 , 69, 216 , 262 , 427, Foias, C. , 7 , 10 , 107 , 143 , 435 431 Fonseca, I. , 107 , 43 5 Costin, O. , 351 , 431 Forrester, P. , 90 , 43 5 Craig, W., 409 , 43 1 Foulquie, A. , 8 , 43 5 Frazho, A. , 7 , 10 , 107 , 143 , 435 Darboux, G. , 24 , 43 1 Fredholm, I. , 10 , 54, 43 5 Daruis, L. , 135 , 431 Freud, G. , 18 , 20, 22 , 24 , 70, 105 , 106 , 130 , Daubechies, L , 38 , 333, 431 132, 134 , 144 , 149 , 151, 160, 171 , 435 Davis, J. , 136 , 43 1 Frobenius, F.G. , 351 , 435 Davis, P. , 105 , 43 1 Frohlich, J. , 299 , 42 7 Day, K. , 313 , 333, 431 Fulton, W. , 351 , 435 Deift, P. , xii, 7 , 21, 68, 177 , 178 , 286, 332, 403, 427 , 431, 432 Gakhov, F. , 313 , 435 de l a Vallee-Poussin, C. , 206 , 43 2 Gamelin, T. , 155 , 435 Delsarte, P. , 8 , 9 , 69 , 70, 90, 105-107 , 212 , Gangbo, W., 107 , 435 216, 43 2 Garcia Lazaro , P. , 107 , 43 5 Delyon, F., xii , 189 , 43 2 Garding, L. , 286 , 43 5 Dembo, A. , 136 , 43 2 Gariepy, R. , 206 , 43 4 Denisov, S. , xii , 20 , 24 , 104 , 105 , 177 , 178 , Garnett, J. , 38 , 155 , 435 197, 203 , 206, 273 , 432 Gasper, G. , 89 , 43 5 Desnanot, P. , 222 , 43 2 Gauss, C.F. , 88 , 43 5 Devinatz, A. , 312 , 331, 375, 43 2 Gautschi, W. , 8 , 43 5 Dewilde, P. , 143 , 432 Gel'fand, I. , 286 , 308, 310, 43 5 Diaconis, P. , 7 , 67 , 68, 348, 351, 352, 430, Genin, Y. , 8 , 9 , 69 , 70 , 90, 105-107 , 212 , 433 216, 43 2 Djrbashian, M. , 9 , 43 3 Geronimo, J. , xi , 10 , 11 , 38, 82, 89, 90, 105 , Dodgson, C.L. , 222 , 43 3 106, 177 , 216 , 231, 238, 261, 293, 344, Doktorskii, R. , 172 , 375, 43 3 411, 434-436 , 44 5 Dombrowski, J. , 285 , 286, 43 3 Geronimus, Ya. , 9-11 , 15 , 24, 39, 69, 70, Donoghue, W. , 239 , 427, 43 3 88-90, 93 , 105-107, 141 , 151, 206, 238, Douglas, J. , 331 , 433 239, 261 , 286, 332, 43 6 Douglas, R. , 313 , 433 Gessel, I. , 7 , 43 6 Dragt, A. , 344 , 43 3 Gesztesy, F., 216 , 293, 428, 431, 436 Dudgeon, D. , 143 , 433 Gioev, D. , 172 , 375, 436 Dunford, N. , 40, 43 3 Glazman, I. , 40 , 277 , 286, 425, 43 6 Durbin, J. , 71 , 433 Godoy, E. , 8 , 107 , 427, 44 5 Duren, P. , 38 , 145 , 43 3 Gohberg, I. , 8 , 40, 55 , 274, 313, 333, 344, Dym, H. , 40 , 143 , 299, 375, 432-434 375, 427 , 43 7 Dyson, F., 351 , 434 Golinskii, B. , 90, 318, 332, 379, 43 7 Golinskii, L. , xi, 6 , 56 , 69, 70, 84, 88-90, Ehrhardt, T. , 332 , 344, 43 4 134, 135 , 151 , 170, 231, 232, 234 , 238, Erdelyi, T. , 11 , 90, 105 , 106 , 43 4 239, 261 , 262, 273, 274, 285-287, 317 , Erdos, P. , 22 , 403, 43 4 410, 437 , 43 8 Evans, L. , 206 , 43 4 Gonchar, A. , 403 , 438 Evans, S. , 351 , 433 Gonzalez-Vera, P. , 9 , 135 , 430, 43 1 Gorodetsky, M. , 333 , 438 Faber, G. , 8 , 43 4 Gragg, W., 261 , 413, 426, 43 8 Falconer, K. , 206 , 43 4 Grenander, U. , 10 , 88, 141 , 155, 170 , 171, Fatou, P. , 38 , 43 4 438 Favard, J. , 11 , 24, 43 4 Grudsky, S. , 6 , 313 , 379, 42 9 Fejer, L. , 37 , 38, 105 , 206, 430, 43 4 Guillemin, V. , 107 , 172 , 375, 429, 43 8 Fekete, M. , 8 , 116 , 134 , 43 4 Feldman, I. , 313 , 43 7 Hahn, W. , 89 , 43 8 Fenchel, W. , 142 , 43 4 Hardy, G. , 55 , 43 8 Feshbach, H. , 299 , 43 4 Harris, J. , 351 , 435 Feynman, R. , 344 , 43 4 Hartman, P. , 346 , 43 8 Finn, J. , 344 , 43 3 Hartwig, R. , 332 , 333, 434 Fischer, E. , 37 , 43 4 Hausdorff, F. , 206 , 43 8 Fisher, M. , 332 , 333, 434 Hayes, M. , 82 , 89, 44 5 AUTHOR INDE X 459

Hayman, W. , 403 , 438 Khrushchev, S. , 9 , 10 , 39, 70 , 89, 98, 107 , Heine, E. , 24 , 43 8 109, 116 , 132 , 156 , 159 , 189 , 197 , Helms, L. , 403 , 438 238-242, 245, 298, 410, 438, 44 2 Helson, H. , 9 , 156 , 43 9 Killip, R. , xii , xiii, 7 , 13 , 142, 143 , 177, 178, Helton, J. , 344 , 43 9 286, 432 , 44 2 Hendriksen, E. , 9 , 43 0 Kirsch, W., 22 , 409, 44 2 Herbert, D. , 409 , 43 9 Kiselev, A., xii , 197 , 216, 286, 431, 436, 44 2 Herglotz, G. , 37 , 38, 43 9 Kolmogorov, A. , 6 , 70 , 71, 141, 170, 171, Hertz, D. , 136 , 43 9 442 Hilbert, D. , 54 , 43 9 Konig, H. , 156 , 42 7 Hille, E., 24 , 43 9 Kotani, S. , xii , 216 , 44 2 Hinton, D. , 216 , 43 9 Krawcewicz, W., 107 , 44 3 Hirschman, I. , 136 , 313, 331, 375, 428, 43 9 Krein, M. , 6-9, 40 , 55, 71, 105, 141 , 155, Hoffman, K. , 38 , 156 , 170 , 43 9 170, 217 , 221, 222, 251, 261, 274, 277 , Hoholdt, T\ , 333 , 439 313, 332 , 344, 425, 426, 429, 437, 44 3 Holden, H. , 216 , 43 1 Kuijlaars, A. , 8 , 43 5 Holland, F. , 155 , 440 Kupin, S. , 177 , 178 , 197, 203, 206, 273, 432, Hopf, E. , 312 , 45 5 443 Howe, R., 344 , 43 9 Howland, J. , 7 , 273, 286, 409, 42 9 Lagrange, J.-L. , 222 , 44 3 Hughes, D. , 136 , 43 9 Laha, R. , 410 , 44 3 Hyam, R. , 221 , 440 Lakaev, S. , 375 , 42 6 Lalesco, T., 55 , 443 Ibragimov, L , 332 , 376, 379, 437, 44 0 Lamperti, J. , 410 , 44 3 Ismail, M. , 7 , 24 , 88-90, 136 , 178 , 428, 430, Landau, H. , xiii , 69 , 105 , 443 440 Landkof, N. , 403, 44 3 Lanford, O. , 142 , 44 3 Jacobi, C, 88 , 222, 44 0 Langer, H. , 9 , 44 3 Jacobson, N. , 344 , 44 0 Laptev, A. , 172 , 178 , 375, 44 3 Jaffard, S. , 333 , 440 Last, Y. , xii , 197 , 206, 286, 287 , 442-44 4 Janssen, A. , 172 , 44 0 Lax, R, 38 , 44 4 Jentzsch, R. , 408 , 44 0 Lebowitz, J. , 351 , 431 Jitomirskaya, S. , 287 , 44 0 Lenard, A. , 332 , 44 4 Johansson, K. , 7 , 68, 351, 352, 368, 376, Lesch, M. , 216 , 44 4 379, 427 , 44 0 Lev-Ari, H. , 9 , 42 5 Johnson, R. , xi , 216 , 261, 293, 409, 411, Levin, E. , 18 , 44 4 435, 436 , 44 0 Levinson, N. , 6 , 70 , 216 , 44 4 Jones, R. , 409 , 43 9 Levitan, B. , 216 , 43 1 Jones, W., 39 , 69, 129 , 135 , 171 , 229, 238, Li, X. , 89 , 44 0 239, 273 , 440, 44 1 Libkind, L. , 171 , 333, 444 Joye, A. , 7 , 273, 286, 409, 429, 44 1 Lidskii, V., 55 , 44 4 Jury, E. , 107 , 441 Littlewood, J. , 55 , 43 8 Justesen, J. , 333 , 439 Lloyd, N. , 107 , 44 4 Lopez, G. , 8 , 286 , 388, 389, 427, 44 4 Kaashoek, M. , 333, 375, 427, 43 7 Lowdenslager, D., 9 , 156 , 43 9 Kac, G. , 287 , 44 1 Lubinsky, D. , 18 , 88, 107 , 189 , 403, 44 4 Kac, M. , 136 , 171 , 331, 333, 368, 375, 441, Lumer, G. , 156 , 44 4 442 Lyons, R. , 155 , 351, 444 Kailath, T. , 6 , 9 , 10 , 71, 143, 313, 425, 429, 432, 442 , 44 6 Macdonald, L , 351 , 444 Kalton, N. , 216 , 43 6 Mackens, W., 136 , 44 4 Kamp, Y. , 8 , 9 , 69, 70 , 90, 105-107 , 212 , Magnus, A. , 189 , 44 4 216, 43 2 Magnus, W., 344 , 44 5 Kato, T. , 55 , 442 Makarov, K. , 216 , 375 , 426, 428, 43 6 Katsnelson, V. , 40 , 43 4 Malamud, M. , 216 , 444, 44 5 Katz, N. , 351, 442 Malaschonok, G. , 39 , 222, 42 6 Kazanjian, N. , 8 , 216 , 45 5 Marcellan, F. , 8 , 24 , 89, 107 , 135 , 238, Kennedy, R, 403 , 43 8 426-428, 431, 435, 444, 44 5 Kesten, H. , 136 , 44 2 Markov, A. , 24 , 221, 445 460 AUTHOR INDE X

Marple, S. , 71 , 143, 445 Pearson, D. , 189 , 44 7 Martinelli, F. , 22 , 409, 44 2 Peherstorfer, F. , 89 , 238, 239, 250 , 44 7 Martinez-Finkelshtein, A. , 8 , 387 , 426, 44 5 Peller, V., 346 , 44 7 Mate, A. , 107 , 134 , 143 , 144 , 151 , 155, 445 Perez, T. , 8 , 44 5 Mathias, R. , 9 , 44 5 Peyriere, J. , 197 , 447 Mattila, R , 206 , 44 5 Phillies, G. , 352 , 44 7 Mazel, D. , 82 , 89, 44 5 Pifiar, M.A. , 8 , 44 5 Mazenko, G. , 352 , 44 5 Pincus, J. , 333 , 430 McCoy, B. , 6 , 117 , 44 5 Pinter, F. , 34 , 70 , 89, 230 , 239 , 261, 438, McLaughlin, K. , 387 , 403, 445, 446 448 Mehta, M. , 7 , 68, 71, 351, 362, 44 6 Plemelj, J. , 313 , 448 Mejlbo, L. , 136 , 44 6 Pollack, A. , 107 , 43 8 Melik-Adamyan, F. , 9 , 44 3 Pollaczek, F., 178 , 44 8 Melman, A. , 136 , 44 6 Polya, G. , 55 , 116 , 438, 44 8 Mersereau, R. , 143 , 433 Meyer, Y., 333 , 440 Rahman, M. , 89 , 43 5 Mhaskar, H. , 403 , 446 Raikov, D. , 308 , 310, 43 5 Miller, P. , 403, 446 Rains, E. , 7 , 42 7 Milnor, J. , 107 , 44 6 Rakhmanov, E. , 9 , 11 , 121, 134, 285, 403, Minami, N. , 413, 446 438, 44 8 Minguez, J. , 8 , 42 8 Raman, S. , 375 , 448 Molchanov, S. , 413 , 446 Ransford, T. , 403 , 448 Moral, L. , xiii , 135 , 239, 262 , 273, 274, 43 0 Rao, R. , 375 , 446, 44 8 Moran, W., 191 , 197, 42 9 Reed, M. , 27 , 40, 44, 45, 47, 50, 51, 55, 261, Moreno-Balcazar, J. , 8 , 427 , 44 5 277, 286 , 44 8 Morf, M. , 143 , 313, 442, 44 6 Reich, E. , 313 , 449 Moser, J. , 409 , 411, 440 Reichel, L. , 413 , 426 Mullikin, T. , 375 , 44 6 Remling, C. , xii , 43 1 Murdock, W. , 136 , 44 2 Rezola, M. , 8 , 42 6 Riesz, F. , 37 , 38, 40, 54 , 160 , 170 , 194 , 197 , Naboko, S. , xii , 178 , 443, 446 449 Nakao, S. , 409 , 44 6 Riesz, M. , 134 , 160 , 170 , 44 9 Natanson, L , 206 , 44 6 Robert, D. , 172 , 375, 443, 449 Nehari, Z. , 345 , 44 6 Robinson, D. , 142 , 44 3 Nevai, P. , xi, xiii, 11 , 18, 22, 24, 34, 56 , 70 , Robinson, E. , 7 , 10 , 44 9 89, 90 , 105-107 , 134 , 143 , 144, 151, Rodman, L. , 216 , 44 9 155, 230-232 , 234 , 238 , 239, 261, 262, Rogers, C. , 206 , 44 9 273, 285 , 287, 386, 434, 438, 445, 446, Rogers, L. , 88 , 44 9 448 Rohatgi, V. , 410 , 44 3 Niewiadomska-Bugaj, M. , 410 , 42 7 Ros, J. , 344 , 44 7 Nikishin, E. , 70 , 216 , 403, 426, 44 6 Rosenblatt, M. , 170 , 171 , 438 Njastad, O. , 9 , 39, 69, 129 , 135 , 171 , 238, Rosenblum, M. , 155 , 44 9 239, 273 , 430, 431, 440, 44 1 Rothe, H. , 88 , 44 9 Novikov, L , 333 , 446 Rovnyak, J. , 155 , 449 Novo, S. , 216 , 44 0 Roy, R. , 88 , 89, 42 6 Nudelman, A. , 9 , 44 7 Rudin, W. , 29 , 37, 132 , 145 , 150 , 158 , 163, Nuttall, J. , 317 , 44 7 164, 333 , 397, 44 9 Rudnick, Z. , 376 , 44 9 Obaya, R. , 216 , 44 0 Ryan, R. , 333 , 440 Okikiolu, K. , 172 , 375, 438, 44 7 Rybalko, A. , 9 , 426, 44 9 Okounkov, A. , 344 , 42 9 Osher, S. , 6 , 44 7 Safarov, Yu. , 172 , 375, 443 Osilenker, B. , 8 , 44 5 Saff, E. , 18 , 88, 105 , 171, 387-389, 403, Oteo, J. , 344 , 44 7 427, 429 , 440, 444-446, 449, 45 0 Safronov, O. , 178 , 443, 450 Pakula, L. , 403 , 447 Sakhnovich, L. , 9 , 216 , 428, 436, 45 0 Pan, K. , 8 , 9 , 435, 44 7 Sansigre, G. , 89 , 44 5 Parter, S. , 6 , 136 , 44 7 Santos-Leon, J. , 135 , 450 Pastur, L. , 409 , 428, 44 7 Sarason, D. , 7 , 45 0 AUTHOR INDE X 461

Sarnak, P. , 351 , 376, 442, 44 9 141, 143 , 144 , 151 , 155, 171, 178, 321, Schatten, R. , 55 , 450 331, 332 , 376, 408, 438, 442, 452, 45 3 Schmidt, P. , 136 , 313, 446, 45 0 Schneider, A. , 216 , 43 9 Ta'asan, S. , 375 , 434 Sch'nol, L , 286 , 45 0 Taylor, S. , 206 , 44 9 Schur, I. , 2 , 10 , 11, 38, 39, 55, 107 , 299, Teicher, H. , 410 , 43 1 317, 351, 435, 45 0 Temme, N. , 90 , 45 3 Schwabl, F., 352 , 45 0 Teplyaev, A. , 9 , 261, 293, 436, 45 3 Schwartz, J. , 40 , 43 3 Thouless, D. , 409 , 45 3 Seiler, E. , 55 , 450 Thron, W. , 39 , 69, 129 , 135 , 171, 229, 238, Semencul, A. , 313 , 437 239, 273, 440, 44 1 Semenov, E. , 333 , 446 Titchmarsh, E.C. , 408 , 453 Serra, S. , 136 , 45 1 Toeplitz, O. , 37 , 38, 45 3 Shahshahani, M. , 351 , 433 Topsoe, F., 142 , 313, 453 Shaw, J. , 216 , 43 9 Totik, V. , 8 , 99 , 105 , 107 , 134 , 143 , 151, Shen, X. , 333 , 454 155, 184 , 189 , 386, 403, 445, 446, Shilov, G. , 308 , 310, 43 5 449-453 Shinbrot, M. , 312 , 43 2 Tracy, C. , 351, 453 Shohat, J. , 24 , 43 9 Trench, W., 136 , 313, 333, 453 Sigal, L , 299 , 42 7 Tsang, T. , 352 , 45 3 Silbermann, B. , 6 , 9 , 142 , 313, 332, 333, Tsekanovskii, E. , 216 , 43 6 379, 429 , 43 4 Tsuji, M. , 403 , 453 Simkani, M. , 403 , 429 Turan, P. , 22 , 97, 107 , 403, 434, 45 3 Simon, B. , xi-xiii, 7 , 11 , 13, 14, 16 , 17 , 20, Ushiroya, N. , xii , 44 2 22, 24 , 25, 27, 29, 40, 44, 45, 47, 50-52, 55, 67 , 68, 99, 104 , 105 , 107 , 115 , 142, Van Assche , W., 8 , 89 , 135 , 189 , 261, 262, 143, 151 , 177, 178 , 184 , 189 , 197 , 216, 273, 431, 438, 444, 45 3 219, 232 , 239, 261, 274, 277 , 285-287, van Moerbeke , P. , 7 , 42 5 293, 332 , 349-351, 409 , 427, 431, 432, van Schagen , F., 333 , 375, 43 7 436, 438, 442, 444, 448, 450, 45 1 Velazquez, L. , xiii , 135 , 239, 262, 273, 274, Sinap, A. , 135 , 453 430 Singer, L , 156 , 427 Verblunsky, S. , xiii, 7 , 10 , 11, 70, 88, 106 , Singh, S. , 317 , 44 7 107, 141 , 217, 221, 222, 238, 45 4 Smirnov, V. , 8 , 40, 151 , 451 Vieira, A. , 143 , 313, 432, 442, 44 6 Sorokin, V., 403 , 446 Vigil, L. , 97 , 107 , 42 6 Soshnikov, A. , 351, 451 von Neumann , J. , 54 , 55, 450, 45 4 Souillard, B. , xii, 189 , 43 2 Voss, H. , 136 , 444, 45 4 Spencer, T. , 285 , 286, 45 1 Spitzer, F., 313 , 376, 450, 45 2 Waadeland, H. , 135 , 171, 441, 454 Spivak, M. , 107 , 45 2 Wall, H.S. , 39 , 69, 45 4 Stahl, H. , 8 , 317 , 403, 452 Walsh, J. , 24 , 43 9 Stanley, R. , 351 , 452 Walter, G. , 333 , 454 Stauffer, D. , 352 , 43 1 Wang, W.-M. , 413 , 454 Steif, J. , 351 , 444 Wendroff, B. , 15 , 24, 45 4 Steinbauer, R. , 89 , 238, 239, 44 7 Wermer, J. , 156 , 402, 403, 454 Steinhardt, A. , 171 , 441 Weyl, H. , 55 , 351, 454 Steklov, V., 121 , 134, 45 2 Widom, H. , 6-9 , 136 , 171 , 172, 313, 332, Stieltjes, T. , 10 , 24 , 251, 452 333, 344 , 351, 375, 379, 403, 428, Stone, M.H. , 24 , 45 2 453-455 Stroock, D. , 410 , 45 2 Wieand, K. , 351 , 455 Sturm, C, 24 , 45 2 Wiener, N. , 6 , 312, 45 5 Suetin, P. , 8 , 45 2 Wilf, EL , 136, 45 5 Sukavanam, N. , 375 , 448 Witte, N. , 7 , 90, 435, 44 0 Sylvester, J. , 39 , 222 , 45 2 Woerdeman, H. , 38 , 43 6 Sz.-Nagy, B. , 40, 44 9 Wojtaszczyk, P. , 333 , 455 Szabados, J. , 107 , 45 2 Wu, J. , 107 , 44 3 Szego, G. , xiv , 2 , 6-11, 24, 26 , 69-71, 88 , 89, 105 , 109, 116 , 121 , 124, 134 , 136 , Youla, D., 8 , 216 , 45 5 462 AUTHOR INDE X

Zeilberger, D. , 222 , 426, 45 5 Zelditch, S. , 172 , 440, 45 5 Zhang, J. , 11 , 90, 105 , 106 , 43 4 Zhou, X. , 403 , 432 Zlatos, A. , 177 , 197 , 45 1 Zygmund, A. , 197 , 45 5 Subject Inde x

absolutely continuou s measure , 4 3 character, 34 9 AKV lemma , 21 7 Chebyshev polynomial s o f the secon d kind , Aleksandrov measure , 35 , 54, 222, 234 , 238, 13 269 Christoffel function , 16 , 117 , 124 , 16 9 Alfaro-Vigil theorem , 9 7 Christoffel-Darboux formula , 18 , 60, 124 , antilinear operator , 4 0 224, 40 3 anti-unitary operator , 4 0 circuit theory , 6 approximate densit y o f zeros, 39 1 CMV basis , 263 , 291 associated polynomial , 24 5 CMV matrix , 287 , 29 3 extended CM V matrix , 29 4 Baker-Campbell-HausdorfT formula , 34 4 CMV representation , 264 , 27 4 balayage, 40 4 alternate CM V representation , 26 4 Banach algebra , 30 8 coefficient stripping , 245 , 25 9 Baxter's lemma , 30 4 commutant liftin g theorems , 7 Baxter's theorem , 4 , 6 , 33, 313 compact operator , 4 5 Bernstein inequality , 12 1 concave, 13 8 Bernstein-Szego approximation , 95 , 122 , concave function , 11 1 143, 148 , 22 5 continued fraction , 20 , 69, 229, 23 5 Bernstein-Szego measure , 111 , 320 Cotes number , 1 7 Bernstein-Szego polynomial , 72 , 8 8 Coulomb energy , 35 5 Beurling algebra , 307 , 32 1 Coulomb gas , 35 2 Beurling weight , 306 , 311, 312, 31 4 Coulomb ga s representation, 6 7 Blaschke product , 25 , 30, 3 6 cyclic vector, 4 2 Blatt-Saff-Simkani lemma , 39 5 Bochner's theorem , 3 8 degree theory , 9 8 Borel transform, 1 2 Deift-Killip theorem , 33 8 Borel-Cantelli lemma , 40 6 Denisov and Kupin' s workshop , 19 7 Borodin-Okounkov formula , 336 , 34 1 density o f states, 2 2 boundary condition , 222 , 259 , 26 9 density o f zeros, 391, 402, 40 4 boundary value , 2 9 derived set , 5 , 4 3 Boyd's theorem , 16 3 determinant, 4 9 determinate, 1 4 canonical decomposition , 4 6 Devinatz's formula , 328 , 339, 345 , 358, 36 3 Cantor set , 19 9 DHK Formula , 37 1 capacity, 40 2 Dirichlet algebra , 15 6 Caratheodory function , 3 , 25, 28, 36, 225, Dirichlet approximation , 20 3 382 discrete spectrum, 4 3 ra-Caratheodory function , 29 4 Dodgson's equality , 21 9 Caratheodory-Toeplitz theorem , 26 , 38, 21 7 doubly substochastic , 4 7 Cauchy inequality , 11 5 Cayley transform , 4 2 eigenvalue, 16 0 CD kernel , 12 4 energy, 39 3 Cesaro approximation , 32 8 entropy, 13 6 Cesaro average , 110 , 40 7 semicontinuity, 13 8

463 464 SUBJECT INDE X equilibrium measure , 40 2 Hilbert-Schmidt operator , 52 , 33 9 essential spectrum , 5 , 24 8 Hilbert-Schmidt theorem , 4 6 essential support, 4 3 Holder continuous , 32 9 Euler's formula , 33 1 Holder's inequality , 5 2 Euler-Wallis formulae , 39 , 6 9 exact dimension , 19 9 Ibragimov's theorem , 321 , 342, 36 8 exact leadin g asymptotics , 9 1 inner function , 3 6 exponential decay , 38 1 inserted mas s point , 7 2 extended CM V matrix , 29 4 inverse Peherstorfer' s formula , 24 7 extreme point , 16 4 inverse Szeg o recursion, 5 9

F. an d M . Ries z theorem , 16 0 J-invariance, 5 8 Favard's theorem , 2 , 14 , 25 1 J-unitarity, 5 8 Fejer approximation , 20 3 Jacobi matrix , 13 , 251 Fejer kernel , 20 6 finite Jacob i matrix , 2 1 Fejer's theorem , 103 , 328 free Jacob i matrix , 1 3 Fejer-Riesz theorem , 26 , 38, 94, 13 5 Jacobi's relatio n o f minors, 22 0 Fenchel's theorem , 14 2 Jensen's inequality , 119 , 143 , 154 , 21 3 Feshbach projection , 29 9 Jentzsch-Szego theorem , 40 9 filtering theory , 6 Kato-Birman theorem , 5 3 finite Jacob i matrix , 2 1 for OPUC , 27 7 finite rank , 29 5 Khrushchev's formula , 287 , 29 8 free Jacob i matrix , 1 3 Khrushchev's workshop , 18 9 Freudian paralle l universe , 132 , 143 , 15 0 Krein algebra , 34 4 FSW duality , 35 0 Krein system , 7 , 9 functional calculus , 4 4 Krein's theorem, 14 1 KW pair , 23 9 Gauss-Jacobi quadrature , 129 , 13 0 Gauss-Jacobi quadratur e formula , 17 , 21 Laurent polynomial , 2 5 Gaussian measure , 7 7 Legendre polynomial , 1 3 Gaussian rando m variable , 34 7 Legendre transform , 14 2 Gel'fand spectrum , 30 8 Levinson algorithm , 6 7 geophysical scattering , 7 Lidskii's theorem , 51 , 55 Geronimus polynomials , 83 , 87, 8 9 linear predictio n theory , 16 7 Geronimus' theorem , 3 , 179 , 219, 226 , 229 , CM. factorization , 26 5 247, 29 8 logmodular algebra , 15 6 Geronimus-Wendroff theorem , 1 5 lower semicontinuous , 39 3 GGT matrix , 25 2 lower triangular, 30 2 GGT representation , 25 2 Lowner order , 4 5 full GG T representation , 25 6 GI approximation , 32 2 m-Caratheodory function , 29 4 GI measure , 32 2 ra-function, 12 , 1 9 Gibbs principle , 14 2 m-Schur function , 29 4 Golinskii's formula , 22 6 mass point, 4 3 Golinskii-Ibragimov theorem , 32 1 inserted mas s point, 7 2 Green's function , 4 1 matrix-valued measure , 206 , 21 2 group representatio n theory , 34 9 matrix-valued polynomial , 8 Mhaskar-Saff theorem , 392 , 41 2 Hankel matrix, 11 , 15, 333 min-max, 4 4 Hankel operator, 333 , 334, 336, 34 4 minimum problem , 120 , 16 5 Hausdorff dimension , 188 , 19 9 mixed C D formula , 22 4 Hausdorff measure , 19 9 modulus, 23 9 Hausdorff-Young inequality , 32 3 moment, 1 1 Heine's formula , 15 , 65 monic orthogona l polynomial , 5 5 Helton-Howe theorem , 340 , 34 2 Herglotz function , 1 2 Nehari's criterion , 34 5 Herglotz representation , 28 , 3 8 Nehari's theorem, 33 4 Hermite polynomial , 1 3 Nevai's conjecture , 17 8 Hessenberg matrix , 252 , 25 4 Nevai-Totik radius , 38 3 SUBJECT INDE X 465

Nevai-Totik theorem , 38 3 m-Schur function , 29 4 Nevanlinna function , 1 2 Schur iterate , 30 , 18 0 nontrivial measure , 1 Schur parameter , 3 , 6, 30 , 21 9 normal operator , 4 0 Schur's recurrenc e relation , 3 1 Schur-Lalesco-Weyl inequalities , 4 8 OPRL, 1 1 second kin d polynomial , 18 , 222, 22 7 orthogonal moni c polynomial , 1 2 second unitarit y condition , 35 0 orthogonal rationa l function , 8 self adjoint operator , 4 0 orthonormal polynomial , 12 , 5 5 sieved polynomial , 84 , 8 9 outer function , 3 7 singular continuou s measure , 4 3 singular inne r function , 3 6 Pade approximant , 229 , 23 8 singular measure , 43 , 15 2 paraorthogonal polynomial , 129 , 130 , 40 7 Sobolev space , 32 9 Peherstorfer's formula , 24 6 spectral measure , 4 2 inverse Peherstorfer' s formula , 24 7 spectral theorem , 4 2 Peherstorfer-Steinbauer theorem , 22 8 spectral theory , 7 Peierls-Bogoliubov inequality , 21 6 spectrum, 4 0 Pick function , 1 2 stationary stochasti c process , 6 Pinter-Nevai formula , 22 9 Steklov conjecture , 12 1 Poisson distribution , 41 3 Stieltjes momen t problem , 14 2 Poisson kernel , 27 , 118 , 151, 404, 41 1 Stieltjes transform , 1 2 Poisson representation , 2 7 strong operato r convergence , 4 1 polar decomposition , 4 6 strong Szeg o theorem, 4 , 321, 348, 368, 37 5 positive operator , 4 5 Sturm compariso n theorem , 2 4 potential, 39 3 Sturm oscillatio n theorem , 2 2 potential theory , 393-39 6 sup-norm algebra , 15 5 pure point , 4 3 symbol, 33 4 Szego asymptotics, 14 4 r-growing, 19 0 Szego condition, 143 , 147, 193 , 256, 272 , Rakhmanov's lemma , 260 , 27 6 314, 319 , 38 2 Rakhmanov's theorem , 5 Szego differenc e equation , 5 6 random matri x theory , 6 Szego function, 109 , 144 , 169 , 173 , 225, rank on e matrix, 29 4 272, 314 , 320, 38 2 rank on e perturbation, 5 3 Szego recurrence, 2 rank tw o perturbation, 29 3 Szego recursion, 56 , 126 , 210, 218, 31 6 ratio asymptotics , 9 1 Szego's theorem, 4 , 109 , 136 , 141 , 154, 155 , rearrangement inequalities , 4 7 158, 163 , 169, 180 , 187 , 212, 35 6 regular point , 9 8 regular value , 9 8 Taylor coefficient , 3 1 relative entropy , 136 , 16 9 three-term recurrence , 6 0 relative Szeg o function, 18 0 three-term recurrenc e relation , 1 2 renormalized determinant , 52 , 55, 27 2 Toeplitz determinant , 4 , 6 , 26 , 109 , 319, reproducing kernel , 16 , 12 0 352 resolvent, 40 , 28 7 Toeplitz matrix , 6 , 26 , 135 , 168, 33 3 restricted densit y o f zeros , 39 1 Toeplitz operator , 9 , 302, 313, 334 reversed polynomial , 2 Totik's workshop , 18 4 Riemann-Hilbert problems , 33 2 trace, 4 9 Riesz product, 189 , 19 1 trace class , 4 9 Rogers-Szego polynomial , 8 7 trace clas s operator, 33 9 Rogers-Szego polynomial , 77 , 82, 8 8 trace ideal , 5 1 root asymptotics , 9 1 trace norm , 27 4 rotation number , 410 , 41 1 transfer matrix , 18 , 22 4 trial function , 280 , 28 4 S-matrix, 34 4 triple product formula , 7 9 Schur algorithm , 30 , 39, 179 , 29 7 Turan measure , 98 , 10 7 Schur approximant , 31 , 35 two-point function , 36 2 Schur basis , 47, 5 1 Schur function , 3 , 6 , 25 , 30, 31, 36, 163, unique representin g measure , 15 5 164, 169 , 235, 248, 297, 298, 31 4 unitary operator , 4 0 466 SUBJECT INDE X unitary operators , 7 upper semicontinuous , 13 8 upper triangular , 30 2

Vandermonde determinant , 68 , 115 , 35 4 variational principle , 13 7 Verblunsky coefficients , 2 , 56, 67, 21 0 periodic Verblunsk y coefficients , 83 , 28 5 Verblunsky formula , 32 , 6 0 Verblunsky's theorem , 2 , 97, 218, 258, 26 8 Verblunsky, Samuel , 22 1 Vitali convergenc e theorem , 38 5

Wall polynomial , 33 , 23 9 weak asymptoti c measure , 40 7 weak operato r convergence , 4 1 Wendroff's theorem , 9 3 Weyl m-function , 23 1 Weyl circle , 23 1 Weyl integratio n formula , 6 8 Weyl solution , 28 8 Weyl's theorem , 5 3 for OPUC , 27 7 WGN disk , 23 1 Widom's formula , 337 , 34 1 Widom's lemma , 39 7 Widom's zer o theorem, 39 7 Wiener algebra , 30 9 Wiener Tauberia n theorem , 30 9 Wiener-Hopf method , 30 3 Wiener-Hopf operator , 303 , 306, 31 0 Wiener-Hopf theorem , 33 6 Wiener-Levy theorem , 310 , 316 , 32 2 wrapped Gaussian , 7 7 Wronskian, 2 4 zeros o f OPRL, 14 , 2 0 zeros o f OPUC, 90 , 398, 40 0 zeros theorem, 90 , 10 2 Titles i n Thi s Serie s

54 Barr y Simon , Orthogona l polynomial s o n the uni t circle , 200 5 53 Henry k Iwanie c an d Emmanue l Kowalski , Analyti c numbe r theory , 200 4 52 Dus a McDuf f an d Dietma r Salamon , J-holomorphi c curve s an d symplecti c topology , 2004 51 Alexande r Beilinso n an d Vladimi r Drinfeld , Chira l algebras , 200 4 50 E . B . Dynkin , Diffusions , superdiffusion s an d partia l differentia l equations , 200 2 49 Vladimi r V . Chepyzho v an d Mar k I . Vishik , Attractor s fo r equation s o f mathematical physics , 200 2 48 Yoa v Benyamin i an d Jora m Lindenstrauss , Geometri c nonlinea r functiona l analysis , Volume 1 , 200 0 47 Yur i I . Manin , Frobeniu s manifolds , quantu m cohomology , an d modul i spaces , 199 9 46 J . Bourgain , Globa l solution s o f nonlinear Schrodinge r equations , 199 9 45 Nichola s M . Kat z an d Pete r Sarnak , Rando m matrices , Frobeniu s eigenvalues , an d monodromy, 199 9 44 Max-Alber t Knus , Alexande r Merkurjev , an d Marku s Rost , Th e boo k o f involutions, 199 8 43 Lui s A . Caffarell i an d Xavie r Cabre" , Full y nonlinea r ellipti c equations , 199 5 42 Victo r Guillemi n an d Shlom o Sternberg , Variation s o n a theme b y Kepler , 199 0 41 Alfre d Tarsk i an d Steve n Givant , A formalization o f set theory without variables , 198 7 40 R . H . Bing , Th e geometri c topolog y o f 3-manifolds, 198 3 39 N . Jacobson , Structur e an d representation s o f Jordan algebras , 196 8 38 O . Ore , Theor y o f graphs, 196 2 37 N . Jacobson , Structur e o f rings, 195 6 36 W . H . Gottschal k an d G . A . Hedlund , Topologica l dynamics , 195 5 35 A . C . Schaeffe r an d D . C . Spencer , Coefficien t region s fo r Schlich t functions , 195 0 34 J . L . Walsh , Th e locatio n o f critical point s o f analytic an d harmoni c functions , 195 0 33 J . F . Ritt , Differentia l algebra , 195 0 32 R . L . Wilder , Topolog y o f manifolds, 194 9 31 E . Hill e an d R . S . Phillips , Functiona l analysi s an d semigroups , 195 7 30 T . Rado , Lengt h an d area , 194 8 29 A . Weil , Foundation s o f algebrai c geometry , 194 6 28 G . T . Whyburn , Analyti c topology , 194 2 27 S . Lefschetz , Algebrai c topology , 194 2 26 N . Levinson , Ga p an d densit y theorems, 194 0 25 Garret t BirkhofF , Lattic e theory , 194 0 24 A . A . Albert , Structur e o f algebras , 193 9 23 G . Szego , Orthogona l polynomials , 193 9 22 C . N . Moore , Summabl e serie s and convergenc e factors , 193 8 21 J . M . Thomas , Differentia l systems , 193 7 20 J . L . Walsh , Interpolatio n an d approximatio n b y rationa l function s i n the comple x domain, 193 5 19 R . E . A . C . Pale y an d N . Wiener , Fourie r transform s i n the comple x domain , 193 4 18 M . Morse , Th e calculu s o f variations i n the large , 193 4 17 J . M . Wedderburn , Lecture s o n matrices , 193 4 16 G . A . Bliss , Algebrai c functions , 193 3 15 M . H . Stone , Linea r transformation s i n Hilber t spac e an d thei r application s t o analysis , 1932 TITLES I N THI S SERIE S

14 J . F . Ritt , Differentia l equation s fro m th e algebrai c standpoint, 193 2 13 R . L . Moore , Foundation s o f point se t theory , 193 2 12 S . Lefschetz , Topology , 193 0 11 D.Jackson , Th e theor y o f approximation, 193 0 10 A . B . Coble , Algebrai c geometr y an d thet a functions , 192 9 9 G . D . Birkhoff , Dynamica l systems , 192 7 8 L . P . Eisenhart , Non-Riemannia n geometry , 192 7 7 E . T . Bell , Algebrai c arithmetic , 192 7 6 G . C . Evans , Th e logarithmi c potential , discontinuou s Dirichle t an d Neuman n problems , 1927 5.1 G . C . Evans , Punctional s an d thei r applications ; selecte d topics , including integra l equations, 191 8 5.2 O . Veblen , Analysi s situs , 192 2 4 L . E . Dickson , O n invariant s an d th e theor y o f number s W. F . Osgood , Topic s i n the theor y o f functions o f several comple x variables , 191 4 3.1 G . A . Bliss , Fundamenta l existenc e theorems, 191 3 3.2 E . Kasner , Differential-geometri c aspect s o f dynamics, 191 3 2 E . H . Moore , Introductio n t o a for m o f general analysi s M. Mason , Selecte d topic s i n the theor y o f boundary valu e problem s o f differentia l equations E. J . Wilczynski , Projectiv e differentia l geometry , 191 0 1 H . S . White , Linea r system s o f curves o n algebraic surface s F. S . Woods , Form s o n noneuclidea n spac e E. B . Va n Vleck , Selecte d topic s i n the theor y o f divergent serie s and o f continue d fractions, 190 5