Orthogona l Polynomial s on the Uni t Circl e Par t 2: Spectra l Theor y This page intentionally left blank http://dx.doi.org/10.1090/coll/054.2
America n Mathematica l Societ y Colloquiu m Publication s Volum e 54 , Par t 2
Orthogona l Polynomial s on the Uni t Circl e Par t 2: Spectra l Theor y
Barr y Simo n
America n Mathematica l Societ y Providence , Rhod e Islan d Editorial Boar d Susan J . Friedlander , 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 . - (America n Mathematica l Societ y colloquiu m 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)— ISBN 0-8218-3675- 7 (par t 2 : alk . paper ) 1. Orthogona l polynomials . I . Title . I L Colloquiu m publication s (America n Mathematica l Society) ; v. 54 .
QA404.5 .S45 200 4 515'.55—dc22 200404621 9
AMS softcover ISBN: 978-0-8218-4864-7 (Vol. 2); set ISBN 978-0-8218-4867-8
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, o r 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 Acquisition s 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 t o [email protected] . © 200 5 by the America n Mathematica l Society . Al l rights reserved . 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-free 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 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 Real Line 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 Verblunsk y Coefficient s an d the Szeg ö Recurrence 5 5 1.6 Example s o f OPUC 7 1 1.7 Zero s and the Firs t Proo f o f Verblunsky's Theore m 9 0 Chapter 2 Szegö' s Theorem 10 9 2.1 Toeplit z Determinant s an d Verblunsk y Coefficient s 10 9 2.2 Extrema l Properties , the ChristofFe l Functions , an d th e Christoffel-Darboux Formul a 11 7 2.3 Entrop y Semicontinuit y an d the First Proo f o f Szegö' s Theorem 13 6 2.4 Th e Szeg ö Function 14 3 2.5 Szegö' s Theorem Usin g the Poisso n Kerne l 15 1 2.6 Khrushchev' s Proo f o f Szegö' s Theorem 15 6 2.7 Consequence s o f Szegö's Theorem 15 9 2.8 A Higher-Order Szeg ö Theorem 17 2 2.9 Th e Relativ e Szeg ö Function 17 8 2.10 Totik' s Worksho p 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 GGT Representatio n 25 1 4.2 Th e CM V Representation 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 CMV Matrice s 29 3 Chapter 5 Baxter' s Theore m 30 1 5.1 Wiener-Hop f Factorizatio n an d the 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 ö 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 an d 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 Secon d Loo k a t 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 Verblunsk y Coefficient s 38 7 Chapter 8 Th e Densit y o f Zero s 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 vi a 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 Rati o Asymptotic s 51 2 9.7 Wea k Asymptotic Measure s 52 1 9.8 Rati o Asymptotics fo r Varyin g Measure s 53 0 9.9 Rakhmanov' s Theore m o n an Ar e 53 5 9.10 Wea k Limits an d Relativ e Szeg ö 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 th e 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, 1) 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 Prüfe r Variable s 66 4 10.13 Modifying th e Measure : Insertin g Eigenvalue s an d Rational Functio n Multiplicatio n 67 3 10.14 Deca y o f CMV Resolvent s 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 Proble m 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 Isospectra l Tor i 78 3 11.9 Quadrati c Irrationalitie s 78 8 11.10 Independenc e o f Spectral Invariants 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 ö Class 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 Rando m Verblunsk y Coefficient s 84 7 12.8 Subshift s 85 5 12.9 Hig h Barriers 86 3 Chapter 1 3 Th e Connectio n t o Jacob i Matrice s 87 1 13.1 Th e Szeg ö Mapping an d Geronimu s Relation s 87 1 13.2 CM V Matrice s an d the Geronimu s Relation s 88 1 13.3 Szegö' 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 Rule and Application s 92 0 13.9 Szegö' 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. Szegö' s theorem 94 6 D. Aleksandro v 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 Verblunsk y 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-Szegö approximatio n 95 5 D. Additional O P formula e 95 6 E. Additional Wal l polynomial formula e 95 6 F. Matri x representation s 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 determinant s (se e also K) 96 1 K. Szegö' 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 to 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 m-functio n 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 Szegö' 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 2
For a n overvie w o f the subjec t o f thes e volumes , se e Sectio n 1. 1 (i n Par t 1) , and fo r a discussion o f how this book came to be and my many debts to others, se e the Prefac e t o Part 1 .
In man y ways , thes e volume s ar e on e larg e boo k broke n i n two , s o I'v e use d successive pag e an d chapte r numbering , startin g i n thi s volum e wher e th e othe r volume lef t off . Th e tabl e o f contents, lis t o f notation, autho r index , an d subjec t index cove r both volumes . I n this volume, there i s a complete bibliography listin g the reference s i n the entir e work . 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 extra 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 will hav e th e fou r edition s tha t Szeg ö did , bu t i t seems likel y ther e wil l b e late r editions . Tha t mean s I especiall y welcom e com - ments, corrections , missin g topic s an d references , an d Informatio n o n ne w pa - pers. 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 tha t you sen d m e attachments . Addend a an d majo r correction s wil l b e poste d a t http://www.math.caltech.edu/opuc.html. M y email i s [email protected] .
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 there 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, ß, 7, T, 5, A, e, C, r\, 0, 6, K, A, A, /Z, U, £, S, TT , II, p, er, E, r, T,
, ^, fi
an Verblunsk y coefficients ; se e (1.1.8 ) ßf Solutio n oiß + ß~l = Ef wit h \ß\ > 1 ; see (13.8.21 ) ß(zo) Jitomirskaya-Las t ratio ; se e (10.10.13 ) dß2n,k{Q) measur e i n varying ratio asymptotics ; se e (9.8.7 ) j(z) Lyapuno v exponent , 7(2: ) = lim™- ^ ^ log ||Tn(z)||; se e (10.5.14 ) ra?A essentia l spectru m fo r Singl e are , T a^\ = {z E <9 D | |arg(Az) | > 2arcsin(a)}; se e (9.9.2 ) re(x) funetio n o f Damanik-Killip , T e(x) = |[M(# ) — v(x)}; se e th e Note s t o Section 13. 1 T0(x) funetio n o f Damanik-Killip , r o(x) = —\[u(x) + v(x)]\ se e th e Note s t o Section 13. 1 T(Bj) contou r surroundin g Bj\ se e (11.7.16 ) T(Gj) contou r surroundin g Gj\ se e (11.7.16 ) Y{z) analyti c funetio n whos e rea l par t i s th e Lyapuno v exponent , 7(2) ; se e (10.11.18) 5QD relativ e Szeg ö funetion; se e (2.9.6 ) A(z) Freu d funetion ; se e Theorem 2.2.1 4 A(z) discriminan t fo r periodi c Verblunsk y 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 ) en{z0) Prüfe r angl e fo r OPUC ; se e (10.12.1 ) Qj CM V building block , O ? - ( *> _?£. ); see (4.2.20 ) n -1 Kn leadin g coefheien t o f
nz + • • •, K n = ||$ n|| ; se e (1.5.1 ) ^oo(C) Christoffe l funetion , Aoo(C ) = inf{/|7r(e 2<9)|2 dfx(0) \ TT a polynomial , TT(C) = 1} ; see (2.2.2 ) Xj(A) eigenvalue s o f a compact Operato r A; se e (1.4.27 ) 26, 2 ^n(C) approximat e Christoffe l funetion , X n(C) = min{J|7r(e )| dfi(0) | deg7r < n, TT(C ) = 1} ; see (2.2.1 ) z(9 p An(£, d/x;p) p-Christoffe l funetion , A n(£,d/x;p) = min{J|7r(e )| d/ji | deg7 r < n , TT(0 = 1} ; see (2.5.1 ) f(x\z) Schu r funetio n o f an Aleksandrov family ; se e Subsection 1.3. 9 xiv NOTATIO N
F(x\z) Caratheodor y functio n o f an Aleksandro v family ; se e (1.3.90 ) <3>^ OP s o f an Aleksandro v family ; se e (3.2.1 ) dji\ 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 Operato r A , i.e., eigenvalue s o f \A\; see (1.4.33 ) dfiac absolutel y continuous part o f the measure d/i, i.e. , i f dp = wff + dps, the n
dpac = t u ff; se e Subsection 1.4. 6 dppp pur e poin t par t o f a measure, dp pp = ^2 X ß({x})ö x; se e Subsection 1.4. 6 dfjis(0) singula r par t o f a measure, dp s = dp — d//ac; se e 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 d/jffl(0) G l approximation s t o a measure; se e (6.1.24 ) i/(n) Beurlin g weight ; se e (5.1.38 ) dv densit y o f zeros measure, dv — limdz/n i f it exists ; se e Section 8. 1 dvn finite densit y o f zeros , dv n — ^ Y^\=i ^ wher e Zj are the zero s o f $ n(z) counting multiplicity ; se e Example 1.7.1 7 dvn,£(0) approximation s use d i n Khrushchev' s theory , dv n^(0) = • (JXp ff ; 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 — Xn -f - Fi^z)\n] se e (4.4.5 ) Uk(Bj) k-th periodi c associate d t o band, Bj\ se e (11.7.16 ) Uk(Gj) k-th periodi c associate d t o gap, G 7; se e (11.7.16 ) Pj Pj = ( 1 - Kf) 1/2; se e (1.5.21 ) p(B) resolven t se t o f an Operato r B, p(B) = {z \ (B — z)~l exists} ; se e Subsec- tion 1.4. 2 p(Tn) Kotani-Ushiroy a ratio ; se e (10.5.83 ) (Td(A) discret e spectrum o f an Operato r A\ se e Subsection 1.4. 6 aess(A) essentia l spectru m o f a n Operato r A, a ess(A) = a(A)\ad(A); se e Subsec - tion 1.4. 6 z n( see an hal f o f CMV basis, a n = \2n = ~ P2n'i (4.2.8 ) cr(B) spectru m o f an Operato r J5 , cr(B) = C\p(B); se e (1.4.2 ) d(J2n{0) measur e use d i n studying varyin g measures ; se e (9.8.4 ) E use d fo r essentia l spectrum o f subshift; se e Theorem 12.8. 1 n+l rn hal f o f CMV basis, rn = X2n- i = z~ ip2n-i\ se e (4.2.7 ) Tn secon d kin d CM V basis; se e (4.4.3 ) (fn normalize d OPUC ; se e (1.5.1 ) (fn Jacob i Solution s at ±2 ; se e (13.1.32 )
#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 wher e 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 aiphabe t fo r subshift ; se e Appendix 10. 5 1 AR annulus , {z \ R' < \z\ < R}; se e (7.1.1) 21 Abe l map; se e (11.7.26) 21 x extende d Abe l map ; se e (11.8.5) 21^ two-side d Beurlin g algebra , {a \ $^n(l + InD^G^ I < oo} ; see (5.1.44) 21+ positiv e Beurlin g algebra , { a E 21^ | a(n) — 0 for n < 0}; see (5.1.44) 2l~ negativ e Beurlin g algebra , {a G 21^ | a(n) = 0 for n > 0}; see (5.1.44) bn Jacob i parameter ; se e (1.2.13) bn^ O P ratio error ; se e (9.1.1) bn(z,d/j,) invers e Schu r iterates ; se e (9.2.14) b^(z) normalize d coefficien t i n quadratic equatio n fo r F; se e (11.7.67) B se t of z's in <9 D for which lim r^i F(rz) doe s not exist; se e (10.10.5) B se t for which 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; se e (1.3.66 ) B(f) rati o D/D; se e (6.2.40 ) B(fi,v) bilinea r potential ; se e (8.1.9 ) B facto r i n CM V proo f o f Geronimus relations ; se e (13.2.16 )
ind cn moment s o f a measure, J e~ dfi(0); se e (1.1.20 ) cn O P hal f o f Solution o f the Geronimo-Cas e 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(ÜT) 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 ; se e (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 5 , wit h ± indicatin g whic h sheet an d i, o whethe r th e contour run s insid e o r outside; se e (11.7.12 ) Cß Canto r se t wit h middl e /3-t h removed; se e Example 2.12. 2 C^ combine d Hausdorf f dimensio n a support ; se e (10.10.6 ) C(f) multiplicatio n Operato r whose Fourier transform actio n is convolution; se e (6.2.10) C(z,w) complet e Cauch y kernel , ^f; se e (1.3.15 ) C th e comple x number s C+ {^GC|Im^>0 } xvi NOTATIO N
Cij(dß) CM V matrix; se e (4.2.12 ) Cij(dfi) 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 ) dj,i 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 Square root integral ; see (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 Fredhol m determinan t o f a trace dass Operator ; se e (1.4.64 ) det2 renormalize d Fredhol m determinan t o f a Hilbert-Schmid t Operator ; se e (1.4.82) ie 2 1 D~<}(e ) th e function i n L (9D, d/x), which i s the boundary valu e o f D(z)~ 1 se t t o zero on a support fo r d/z s; se e (2.4.33 ) Dn(d/jJ) 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 ö function, exp( / f£± f \og(w(6))^); se e (2.4.2 ) D^(zo) loca l infinitesimal Hausdorf f dimension ; se e (10.8.28 ) D uni t disk , {z \ \z\ < 1 } BR {z | \z\ < R] D°°,c se t o f Schur parameters; see Subsection 1.3. 6 V({Gj}) Dirichle t dat a torus ; se e Section 11. 4
Ep 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 ) se e £o(J) energ y term i n C 0 su m rule , £Q{J) = Ylj,± l°g\ßf(J)\'i (13.8.52 ) £q(ß) periodize d CM V matrix; se e (11.2.4 ) £(d/jb) Coulom b energy fo r potential theory £{djj) = J log\z — w\~l dfi(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) f-(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 ) f^(z) Schu r approximants ; se e (1.3.41 ) f(k) Jos t functio n fo r Schrödinge r Operators ; se e (10.7.29 ) f(x, k) Jos t Solutio n fo r Schrödinge r Operators ; se e (10.7.26 ) f(z) Schu r functio n fo r a measure ^ zJf\zl — f ~w~z^ d^O); se e Sectio n 1. 3 2 1 2 F functio n i n P 2 su m rule ; fo r E > 2 , F(E) = \ J^^E - 4] / dE; se e (13.8.27) an d (13.8.33 ) Fn(f) Feje r approximation ; se e (2.12.33 ) FW 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 fi^r f dß(ß)\ se e Section 1. 3 NOTATION xvi i
1 n F(dß) leadin g term i n limit o f Toeplitz matrices, F(dß) — lim^^oo D n(d/j/) ^ — njlot1 - l ail2); see Theore m 2.1. 2 ie i9 T Fourier-CM V transform , (Tf) n = J Xn(e ) f(e ) dfi(0); se e (10.7.79 ) To fre e Fourier-CM V transform , T fo r d/ x = |£ ; se e (10.7.79 ) Tu{d\i) fül l GG T matrix ; se e (4.1.25 ) an d Propositio n 4.1. 3 g matri x Schu r function ; se e (4.5.13 ) #o 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,dfj) functio n neede d i n Khrushchev theory ; se e (9.2.32 ) G auxiliar y functio n fo r P2 sum rul e give n b y G(a) = a 2 — 1 — log(a2); se e (13.8.28) Goo limi t valu e fo r Geronimo-Cas e equations ; se e (13.6.43 ) Go(z) unperturbe d matri x Caratheodor y function ; se e (4.5.12 ) 2 Ga,\(z) allowe d limi t fo r rati o asymptotics , G a,x(z) = |[( 1 + Xz) + [( 1 — Xz) - f 4a2 As]1/2]; se e (9.5.5 ) Gj gap s i n 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) + *nW; se e (3.2.40) . Up to a constant, thi s i s the Uk of (9.2.28) . se e Gn(z) forma l dua l functio n t o Golinskii' s G n, G n(z) = F(z)$ n - ^ n{z)\ (3.2.41). U p to a constant, thi s i s the u* k of (9.2.28) . G(d/j) secon d ter m i n Toeplit z determinan t asymptotics , G(d/u,) = n^=o( l — 2 1 |aj| )^- ;see (2.1.3 ) G(z) matri x Caratheodor y function , G(z) = P[y±^]P; se e (4.5.11 ) GW J A se e (10.1.5) G(z) Green' s functio n fo r £; se e (10.11.25) C/oo noneigenvalu e set ; see Theorem 10.1. 5 gki({an}%L0) GG T matrix ; se e (4.1.4 ) an d (4.1.5 ) Gn MN T integra l Operator ; se e (9.4.28 ) ha(A) Hausdorf f a-dimensiona l measure ; se e (2.12.8 ) fix Hanke l determinant; se e (1.2.29 ) h(a) Hanke l Operator ; se e (6.2.2 ) h(ä) Hanke l alternate Operator ; se e (6.2.2 ) 1 2 1 2 2 2 H ' Sobole v space , H ' = { / G L (ÖD, §) | £|fc||/fc| < °o} ; 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 ) 'Hac L 2(9D, d/Xac); see Subsection 1.4. 7 %pp spac e o f eigenvectors; se e Subsection 1.4. 7 2 Hsc L (öB,d/xsc); se e Subsection 1.4. 7
/(/) log(D/D); se e (6.2.39 ) X\ trac e dass, {^ 4 | Tr(|^4|) < oc} ; see Subsection 1.4.1 2 X2 Hilbert-Schmid t ideal , {A | Tr(AM) < 00} ; see Subsectio n 1.4.1 3 Z^ invarian t measures on dB fo r a measure \x on U(l, 1); see Proposition 10.6. 2 Xp trac e ideal , {, 4 | Tr(|^4|p) < 00} ; see Subsection 1.4.1 3 xviii NOTATIO N
X(A) se t o f measures o n dB lef t invarian t b y the protection actio n o f A; see Theorem 10.4.1 2
J J = (J _?:), relevant t o the definition o f U(l, 1); see (10.4.1) J^1) onc e stripped 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 <£ ^ 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; se e (11.7.25 ) _1 Je restrictio n o f S(C + C )
Kn normalizatio n facto r o f 3> ^ in th e theor y o f matrix-valued measures ; se e (2.13.22) Kn(z,() Christoffel-Darbou x kernel ; see (1.2.36 ) an d (2.2.17 ) LA induce d projectiv e actio n o f A o n <9B ; see (10.4.26 ) irne Lm Fourie r coefficient s o f log(w), i.e., L m = J e~ log(w(0))ff ; se e (1.1.23 ) C perio d lattic e o f the Rieman n surfac e see (4.2.17 ) Cq periodize d C matrix; se e (11.2.7 ) C(ÜÜO) lef t limi t point s o f {T nwo}\ se e (10.5A.1 ) (•, -)L scala r inne r produc t o n matrix-valued functions ; se e Section 1. 1 ((•, • ))L matri x inne r produc t o n matrix-valued functions ; se e (2.13.3 ) m+ m-functio n use d in Kotani theory, m+ = pQ lz(l — äo/); se e (10.11.5 ) an d Theorem 10.11. 6 m*(xi,...,Xj) functio n i n DH K formula , m*(#i,... Xj) = max(0, x\,x\ - f x 2) • • •, x\ + h Xj); se e (6.5.15 ) mß(E) Wey l m-function fo r Jacob i matrices , m ß(E) = J x _^; se e (1.2.6 ) n 2 Mi/2(d/x) G l siz e on measures, M 1/2{dfji) = J2™=i l^n(d/i)| ; se e (6.1.16 ) M^j matri x invers e to matrix-valued Toeplit z matrix ; se e (2.13.14 ) p M(a0,..., otp-i) modulu s o f a point i n B ; see (11.4.29 ) Mf reflectio n ma p associate d t o CMV , M, (Mf)(z) = /(*); see (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 + ä0)(l - F(z)); se e (11.7.76 ) M+,i(X) probabilit y measure s o n a compact Hausdorf f space , X Mij(dp) M hal f o f CM V CM factorization , Mij = {x^Xj)\ se e (4.2.16 ) Mq(ß) periodize d M matrix ; se e (11.2.7 ) I6 1 2 l0 t ie NA(Z ) nor m associated to a projective actio n o f A, i f UQ — 2~ / (e ) , N a(e ) = \\Aue\\] se e (10.4.30 ) 0(f] ZQ) degre e o f a meromorphic function ; se e (11.7.11 ) n ( ö(u>o) orbi t o f a dynamical System , O(LJ 0) = {T üü} ^=_00; se e (10.5A.1 ) pn normalize d OPRL ; se e (1.2.5 ) NOTATION xi x pn decayin g CM V function, p n = yn + F{z)x n; se e (4.4.4 ) P pur e poin t support , U x{z \ fi\({z}) ^ 0 } for an Aleksandro v family ; se e (10.10.1) P+ fundamenta l projectio n i n Wiener-Hopf theory ; see 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) se e 3 14 Pr(6, n sn hal f o f alternate CM V basis, s n = x^n — z~ (p2n] se e (4.2.10 ) sn(f) n ~th Taylor coefficien t o f /; se e (1.3.42 ) s(fi) entrop y o f \x in Furstenberg's theorem ; see (10.6.10 ) S combine d singula r support s o f an Aleksandrov family ; see (10.10.2 ) Sa,s(Ä) Hausdorf f measur e constructor ; se e (2.12.7 ) 5A suppor t o f singular par t o f dß\\ se e (10.1.2 ) S(a) matri x use d i n the Killip-Nenci u proo f o f the Geronimu s relations ; se e (13.2.13) S(fi I v) relativ e entropy ; se e (2.3.2) n S(z) S(z) = £~=0 *nz ; se e (7.2.3 ) Sz(£) Szeg ö transform o f measures; se e (13.1.4 ) SIL(2, C) 2x 2 comple x matrice s o f determinant 1 xx NOTATIO N SL(2,R) 2x 2 rea l matrices o f determinant 1 SU(1,1) A e SL(2 , C) s o that A* JA = J; se e (10.4.1 ) SU(1,1; Jr) A e 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 o f the Geronimus connection; see (13.2.14) TOQ(Z) limi t o f modified transfe r matrice s wit h a n £ t}\ se e (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 ) ^tu...,tk(g) multicharacte r fo r SU(n); see (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 dass; 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 hal f o f decayin g Solutio n o f transfe r matrix , Uk = ^k + F(z)ipk\ se e (9.2.28) u*k botto m hal f o f decaying Solutio n o f transfer matrix , u^ = —ip^ + F(z)(p^; see (9.2.28 ) un rati o use d i n inverse Geronimu s relations , u n — ^n+2/^n+i5 se e (13.1.34 ) and (13.1.37 ) un(z, J) Jos t functio n fo r OPRL ; see Section 13. 6 U^(x) potentia l o f a measure /ionC; se e (8.1.8 ) U(P, Q) ma p i n Kato similarity transforms , U(P, Q) — YJJ=I QjPj\ se e (12.1.10 ) U(l, 1 ) Ae GL(2 , bbC) so Ä* = JA = J; se e Section 10. 4 U(n) n x n unitar y matrice s ( : see vn rati o used in inverse Geronimus relations, v n = —(Pn+2/ Pn+i > (13.1.34 ) and (13.1.37 ) Vk canonica l basis o f first kin d holomorphic differential s o n the Riemann sur - face 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) contribution s has to deal with the issu e illustrated b y the fac t tha t i n their Englis h translations , th e nam e 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 th e reader i s warned that thi s may produc e a difficult y i f yo u ar e tryin g t o orde r a book o n Amazo n usin g a n autho r spellin g from thi s bibliography . M.J. Ablowitz , D.J . Kaup , A.C . Newell , an d H . Segur , The inverse scattering transform- Fourier analysis for nonlinear problems, Studie s in Appl. Math. 53 (1974) , 249-315. (Cite d on 782. ) M.J. Ablowit z an d J.F . Ladik , Nonlinear differential-difference equations, J . Math . Phys . 16 (1975) , 598-603. (Cite d o n 782 , 808. ) M.J. Ablowitz and J.F. Ladik, Nonlinear differential-difference equations and Fourier anal- ysis, J . Math . Phys . 1 7 (1976) , 1011-1018 . (Cite d o n 782 , 808. ) M.J. Ablowit z an d J.F. Ladik , A nonlinear difference scheme and inverse scattering, Stud - ies i n Appl. Math . 5 5 (1976) , 213-229. (Cite d o n 782 , 808. ) M.J. Ablowit z an d J.F . Ladik , On the Solution of a dass of nonlinear partial difference equations, Studie s i n Appl . Math . 5 7 (1976/77) , 1-12 . (Cite d o n 782. ) A.V. Abramyan , On circular parameters, Mosco w Univ . Math . Bull . 4 2 (1987) , 60-62 ; Russian origina l i n Vestnik Moslov . Univ . Ser . I Mat. Mekh . 10 4 (1987) , 73-75. (Cite d o n 141, 332. ) R. Ackner , H . Lev-Ari , an d T . Kailath , The Schur algorithm for matrix-valued meromor- phic functions, SIA M J . Matri x Anal . Appl . 1 5 (1994) , 140-150 . (Cite d o n 9. ) M. Adler and P . van Moerbeke, Integrals over classical groups, random permutations, Toda and Toeplitz lattices, Comm . Pur e Appl . Math . 5 4 (2001) , 153-205 . (Cite d o n 7. ) M. Adle r an d P . va n Moerbeke , Recursion relations for unitary integrals, combinatorics and the Toeplitz lattice, Comm . Math . Phys . 23 7 (2003) , 397-440. (Cite d o n 7. ) J. Agie r an d J.E . 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R. , 24 , 107 , 101 3 Berezin, F., 136 , 98 7 Ambarzumian, V. , 815 , 98 4 Berg, C. , 136 , 98 7 Ambroladze, A. , 24 , 98 4 Berg, L. , 333 , 988 Amdeberhan, T. , 222 , 98 4 Bernstein, S. , 88 , 134 , 98 8 Ammar, G. , 261 , 413, 98 5 Berriochoa, E. , 8 , 134 , 171, 881, 888, 889, Anderson, R, 847 , 98 5 985, 98 8 Beurling, A. , 40 , 313, 98 8 Andrews, G. , 88 , 89, 98 5 Billingsley, R, 588 , 98 8 Andrievskii, V. , 403 , 98 5 Birman, M.S. , 55, 261, 277, 700, 98 8 Aptekarev, A. , 8 , 171 , 216, 741 , 985 Birnir, B. , 782 , 98 8 Area, L , 8 , 98 5 Blanchard, J. , 107 , 1007 , 100 8 Arens, R. , 156 , 98 5 Blaschke, W., 40 , 98 8 Armitage, D. , 741 , 985 Blatt, H.-R , 403 , 985, 98 8 Arnol'd, V.l. , 808 , 98 5 Blatter, G. , 7 , 273, 988 Aronszajn, N. , 239 , 550 , 98 5 Bloch, F., 723 , 98 8 Askey, R. , 88-90 , 134 , 98 5 Bobenko, A. , 782 , 98 7 Atkinson, F.V. , xiii , 69 , 98 5 Bocher, M. , 25 , 98 8 Aubry, S. , 863 , 98 6 Bochner, S. , 156 , 98 8 Avila, A. , 707 , 98 6 Boole, G. , 868 , 98 8 Avron, J. , 22 , 409, 605, 706, 707 , 825, 98 6 Borg, G. , 815 , 824, 98 8 Borodin, A. , 344 , 98 8 Bach, V., 299 , 98 6 Böttcher, A. , 6 , 9 , 116 , 142 , 313, 331-333, Badkov, V. , 89 , 90, 151 , 986 344, 375 , 379, 98 9 Baik, J. , 7 , 544, 685, 98 6 Bougerol, R, 616 , 98 9 Bakonyi, M. , 61, 69, 216 , 262 , 98 6 Bourgain, J. , 707 , 98 9 Balslev, E. , 690 , 98 6 Bourget, O. , 7 , 273, 286, 409, 706, 724 , 847, Barbey, K. , 156 , 98 6 989 Bargmann, V. , 684 , 700 , 98 6 Boutet d e Monvel , L. , 172 , 98 9 1032 AUTHOR INDE X Bovier, A. , 863 , 989 Davis, R , 105 , 993 Boyd, D. , 170 , 98 9 Day, K. , 313 , 333, 99 3 Bressoud, D. , 222 , 98 9 De Concini , C, 782 , 788 , 99 3 Brezinski, C, 69 , 98 9 Deift, R , xii , 7 , 21, 68, 177 , 178 , 286, 332 , Browder, A. , 155 , 989 403, 544 , 663, 685, 701, 936, 937 , 986 , Browder, F. , 286 , 98 9 993 Brown, A. , 801 , 989 de l a Calle , B. , 534 , 99 3 Brown, G. , 191 , 197, 98 9 de l a Vallee-Poussin, C, 206 , 816 , 99 3 Browne, D. , 7 , 273, 988 del Rio , R. , 550 , 559, 563, 639, 652, 869 , Bruckstein, A. , 6 , 98 9 994 Bulla, W. , 782 , 99 0 Delsarte, R, 8 , 9 , 69 , 70 , 90, 105-107 , 212 , Bultheel, A. , 9 , 99 0 216, 978 , 99 4 Bump, D. , 67 , 68, 348, 351, 990 Delyon, F. , xii , 189 , 558, 855, 99 4 Buys, M. , 782 , 99 0 Dembo, A. , 136 , 99 4 Denisov, S. , xii , 20 , 24 , 104 , 105 , 177 , 178 , Cachafeiro, A. , 8 , 134 , 171 , 684, 798 , 881, 197, 203 , 206, 273, 474, 475 , 631, 892, 888, 889 , 985, 988, 99 0 895, 936 , 937 , 99 4 Calvetti, D. , 413 , 985 Desnanot, R, 222 , 99 4 Cantero, M. , xiii , 135 , 239, 262 , 273, 274, Devinatz, A. , 312 , 331, 375, 994, 99 5 975, 979 , 99 0 Dewilde, R, 143 , 995 Caratheodory, C, 37 , 38, 990, 99 1 Diaconis, R, 7 , 67, 68 , 348, 351, 352, 990 , Carey, R. , 333 , 991 995 Carleman, T. , 55 , 99 1 Djrbashian, M. , 9 , 99 5 Carmona, R. , 605 , 616, 617, 630 , 706 , 847, Dodgson, C.L. , 222 , 99 5 972, 99 1 Doktorskii, R. , 172 , 375, 99 5 Carr, J. , 808 , 101 5 Dombrowski, J. , 285 , 286, 824 , 825 , 99 5 Case, K.M. , xiii , 10 , 177 , 344, 630, 631, Donoghue, W. , 239 , 550 , 799 , 985, 99 5 903-905, 911 , 937, 991, 999 Douglas, J. , 331 , 995 Cesari, L. , 824 , 99 1 Douglas, R. , 313 , 99 5 Chadan, K. , 630 , 99 1 Dragt, A. , 344 , 99 6 Chams, J. , 178 , 99 1 Dubrovin, B. , 782 , 807 , 99 6 Chebyshev, R, 24 , 99 1 Dudgeon, D. , 143 , 99 6 Chen, Y. , 90 , 136 , 333, 987 Dunford, N. , 40, 685, 99 6 Chihara, T. , 24 , 99 1 Durbin, J. , 71 , 996 Choquet, G. , 605 , 99 1 Düren, R, 38 , 145 , 556, 936 , 99 6 Chow, Y.S., 410 , 99 1 Dym, H. , 40 , 143 , 299, 375 , 995, 99 6 Chowdhury, D. , 352 , 99 1 Dyson, F. , 351 , 996 Christ, M. , xii , 673 , 991 Christoffel, E. , 24 , 684, 99 1 Eastham, M.S.R , 718 , 723, 99 6 Chulaevsky, V. , 707 , 99 1 Eggarter, T. , 672 , 99 6 Clark, S. , 216 , 815, 991, 992 Ehrhardt, T. , 332 , 344 , 996 , 99 7 Cohn, A. , 107 , 99 2 Eilbeck, J. , 808 , 101 5 Combes, J.M. , 299 , 690, 983, 986, 99 2 Eliasson, L. , 707 , 99 7 Constantinescu, T. , 61 , 69, 216 , 262 , 986, Enol'skii, V. , 782 , 98 7 992 Ercolani, N. , 782 , 808, 997, 101 5 Costin, O. , 351 , 992 Erdelyi, T. , 11 , 90, 105 , 106 , 978, 99 7 Craig, W. , 409 , 605 , 99 2 Erdos, R, 22 , 403, 99 7 Cwikel, M, 700 , 99 2 Evans, L. , 206 , 99 7 Cycon, H, , 605 , 99 2 Evans, S. , 351 , 995 Damanik, D. , 630 , 663, 672, 700 , 862, 863, Faber, G. , 8 , 99 7 881, 899 , 903, 911, 912, 915, 918, 920, Faddeev, L. , 937 , 102 9 938, 944 , 99 2 Falconer, K. , 206 , 99 7 Darboux, G. , 24 , 99 2 Farkas, H. , 741 , 763, 782, 99 7 Daruis, L. , 135 , 99 3 Fatou, R , 38 , 99 7 Date, E. , 782 , 99 3 Favard, J. , 11 , 24, 99 7 Daubechies, L , 38 , 333, 993 Fay, J. , 783 , 788, 99 7 Davies, E.B. , 544 , 663, 99 3 Fejer, L. , 37 , 38, 105 , 206, 991, 997 Davis, J. , 136 , 99 3 Fekete, M. , 8 , 116 , 134 , 99 7 AUTHOR INDE X 1033 Feldman, L , 313 , 100 1 Giachetti, R. , 782 , 100 0 Fenchel, W., 142 , 99 7 Gilbert, D. , 638 , 639, 647, 706 , 1000 , 100 1 Feshbach, H. , 299 , 99 7 Gioev, D. , 172 , 375, 100 1 Feynman, R. , 344 , 99 7 Glazman, I. , 40 , 277 , 286, 685, 984, 100 1 Figotin, A. , 605 , 672, 847, 101 8 Godoy, E. , 8 , 107 , 684, 985, 1001 , 101 3 Finkel, A. , 782 , 99 0 Gohberg, L , 8 , 40, 55 , 274, 313, 333, 344, Finn, J. , 344 , 99 6 375, 987 , 100 1 Fischer, E. , 37 , 99 7 Goldberg, W. , 782 , 100 1 Fisher, M. , 332 , 333, 997 Gol'dsheid, L , 616 , 100 1 Flaschka, H. , 782 , 807 , 808, 816, 937, 973, Goldstein, M. , 707 , 989, 100 1 997, 99 8 Golinskii, B. , 90 , 318, 332, 379, 100 1 Floquet, G. , 723 , 998 Golinskii, L. , xi, 6 , 56 , 69, 70 , 84 , 88-90, Foias, C. , 7 , 10 , 107 , 143 , 485, 99 8 134, 135 , 151, 170, 231, 232, 234, 238, Fonseca, L , 107 , 99 8 239, 261 , 262, 273, 274, 285-287, 317, Forrester, R, 90 , 99 8 410, 492 , 529, 544 , 550 , 558, 605, 630, Foulquie, A. , 8 , 99 8 639, 825 , 829, 834 , 845 , 847, 975, 978, Frazho, A. , 7 , 10 , 107 , 143 , 485, 99 8 1001, 100 2 Fredholm, L , 10 , 54, 99 8 Golubitsky, M. , 801 , 100 2 Freud, G. , 18 , 20, 22, 24, 70 , 105 , 106 , 130 , Gonchar, A. , 403 , 937, 100 2 132, 134 , 144 , 149 , 151 , 160, 171 , 998 Gonzalez-Vera, R, 9 , 135 , 990, 99 3 Frobenius, F.G. , 351 , 998 Gordon, A. , 559 , 563, 707, 100 2 Froese, R. , 605 , 99 2 Gorodetsky, M. , 333 , 100 2 Fröhlich, J. , 299 , 707 , 986, 99 8 Gragg, W. , 261 , 413, 985, 100 2 Fulton, T. , 685 , 99 8 Gredeskul, S. , 672 , 100 3 Fulton, W. , 351 , 998 Greene, J. , 937 , 99 8 Furstenberg, H. , 605 , 616, 617, 99 8 Grenander, U. , 10 , 88, 141 , 155, 170 , 171, 1003 Gakhov, F. , 313 , 99 8 Griffiths, R , 741 , 763, 782, 100 3 Gamelin, T. , 155 , 99 8 Grudsky, S. , 6 , 313 , 379, 98 9 Gangbo, W. , 107 , 99 8 Guillemin, V. , 107 , 172 , 375, 801, 989, Garcia-Amor, J. , 881 , 888, 889, 98 8 1002, 100 3 Garcia Lazaro , R, 107 , 99 8 Guillot, M. , 534 , 100 3 Gardiner, S. , 741 , 985 Guivarc'h, Y. , 616 , 100 3 Garding, L. , 286 , 99 8 Gardner, C. , 782 , 807, 937 , 998, 101 5 Hahn, W. , 89 , 100 3 Gariepy, R. , 206 , 99 7 Hamel, G. , 718 , 100 3 Garnett, J. , 38 , 155 , 733, 741, 799, 99 8 Hardy, G. , 55 , 100 3 Gasper, G. , 89 , 99 9 Harris, J. , 351 , 741, 763, 782, 998, 100 3 Gauss, C.F. , 88 , 99 9 Hartman, R , 346 , 100 3 Gautschi, W. , 8 , 521, 999 Hartwig, R. , 332 , 333, 99 7 Gel'fand, L , 286 , 308, 310, 684 , 724 , 99 9 Haupt, O. , 718 , 100 3 Gel'fond, A. , 520 , 99 9 Hausdorff, F. , 206 , 100 3 Genin, Y. , 8 , 9 , 69 , 70 , 90, 105-107 , 212 , Hayes, M. , 82 , 89, 101 5 216, 978 , 99 4 Hayman, W., 403 , 100 3 Germinet, F. , 707 , 99 9 Hedlund, G. , 605 , 101 6 Geronimo, J. , xi , 10 , 11, 38, 82, 89, 90, 105, Heine, E. , 24 , 100 3 106, 177 , 216, 231, 238, 261, 293, 344, Helms, L. , 403 , 100 3 411, 485 , 605, 630, 631, 663, 709, 741, Helson, H. , 9 , 156 , 100 4 782, 783 , 788, 903-905, 911, 920, 978, Heiton, J. , 344 , 100 4 997, 999 , 101 5 Hendriksen, E. , 9 , 99 0 Geronimus, Ya. , 9-11 , 15 , 24, 39, 69, 70, Henon, M. , 801 , 807, 100 4 88-90, 93 , 105-107, 141 , 151, 206, 238, Herbert, D. , 409 , 100 4 239, 261, 286, 332 , 707 , 718, 729, 880, Herglotz, G. , 37 , 38, 100 4 881, 975 , 977, 978 , 999, 100 0 Herman, M. , 707 , 100 4 Gessel, L , 7 , 100 0 Hernandez, R. , 534 , 1003 , 100 4 Gesztesy, F., 216 , 293, 558, 684, 782 , 783, Hertz, D. , 136 , 100 4 815, 816 , 987 , 990-992, 999 , 100 0 Hubert, D. , 54 , 100 4 Ghez, J.-M. , 863 , 98 9 Hill, G. , 718 , 100 4 1034 AUTHOR INDE X Hille, E. , 24 , 100 4 Kamp, Y. , 8 , 9 , 69 , 70 , 90, 105-107 , 212 , Hinton, D. , 216 , 100 4 216, 978 , 99 4 Hirschman, I. , 136 , 313, 331, 375, 987, Kato, T. , 55 , 517, 691, 714, 825 , 100 8 1004, 100 5 Katsnelson, V. , 40 , 581, 996, 100 8 Hochstadt, H. , 723 , 782, 815, 816, 100 5 Katz, N. , 351, 100 8 Hoffman, K. , 38 , 156 , 170 , 100 5 Katznelson, Y. , 554 , 605, 100 9 Hoholdt, T. , 333 , 100 5 Kaup, D. , 782 , 98 3 Holden, H. , 216 , 782 , 783, 815, 990, 992, Kazanjian, N. , 8 , 216 , 102 8 999, 100 0 Kennedy, R, 403 , 100 3 Holland, F. , 155 , 100 5 Kesten, H. , 136 , 605, 998, 100 9 Hopf, E. , 312 , 102 8 Khan, S. , 647 , 100 9 Howe, R. , 344 , 100 4 Khinchin, A. , 605 , 100 9 Howland, J. , 7 , 273, 286, 409, 706 , 724 , Khrushchev, S. , 9 , 10 , 39, 70 , 89 , 98, 107 , 847, 98 9 109, 116 , 132 , 156 , 159 , 189 , 197 , Hughes, D. , 136 , 100 5 238-242, 245 , 298, 410, 485, 486, 492, Hundertmark, D. , 700 , 701, 899, 931, 937, 503, 511 , 523, 524, 529, 792 , 798 , 975, 992, 100 5 978, 1002 , 100 9 Hyam, R. , 221 , 100 5 Killip, R. , xii , xiii, 7 , 13 , 142 , 143 , 177, 178 , 286, 624 , 638, 663, 672, 700 , 863, 881, Ibragimov, L , 332 , 376, 379, 1001 , 100 5 888, 899 , 903, 925, 931, 936, 937 , 992 , Iochum, B. , 863, 98 7 993, 100 9 Ishii, K. , 581 , 605, 100 5 Kingman, J. , 605 , 100 9 Ismail, M. , 7 , 24 , 88-90, 136 , 178 , 684, 987, Kirsch, W., 22 , 409, 605, 992, 100 9 991, 100 5 Kiselev, A., xii , 197 , 216, 286 , 606, 638, Israel, R. , 605 , 100 5 651, 652 , 668, 672, 673, 826, 845 , 855, Its, A. , 782 , 987 , 100 6 991, 1000 , 100 9 Kohmoto, M. , 863 , 100 9 Jacobi, C, 88 , 222, 100 6 Kohn, W. , 684 , 100 7 Jacobson, N. , 344 , 580 , 100 6 Kolmogorov, A. , 6 , 70 , 71, 141, 170, 171, Jaffard, S. , 333 , 100 6 1009 Janssen, A. , 172 , 100 6 Konig, H. , 156 , 98 6 Javrjan, V. , 557 , 100 6 Kooman, R. , 824 , 100 9 Jentzsch, R. , 408 , 100 6 Koranyi, A. , 799 , 100 9 Jitomirskaya, S. , 287 , 563, 631, 638, 639, Kotani, S. , xii , 216 , 558 , 598, 606, 647, 663, 647, 707 , 868, 869, 994, 999, 1002 , 100 6 855, 101 0 Johansson, K. , 7 , 68, 351, 352, 368, 376, Kra, L , 741 , 763, 782, 99 7 379, 986 , 100 6 Kramers, H. , 718 , 101 0 Johnson, R. , xi , 216 , 261, 293, 409, 411, Krasovsky, L , 707 , 100 6 663, 672 , 709 , 741, 782, 783, 788, 978, Krawcewicz, W., 107 , 101 0 993, 999, 1000, 1006 Krein, M. , 6-9 , 40 , 55 , 71, 105, 141, 155, Jones, R., 409, 1004 170, 217 , 221, 222, 251, 261, 274, 277 , Jones, W., 39, 69, 129, 135, 171, 229, 238, 313, 332 , 344, 684, 984, 988, 1001 , 101 0 239, 273,1006,100 7 Krengel, U. , 588 , 101 0 Jost, R. , 684 , 100 7 Krichever, I. , 782 , 808, 996, 1010 , 101 5 Joye, A. , 7 , 273, 286, 409, 706 , 724 , 847, Krikorian, R. , 707 , 98 6 989, 100 7 Kruskal, M. , 782 , 807, 937, 998, 101 5 Jury, E. , 107 , 1007 , 100 8 Kuijlaars, A. , 8 , 99 9 Justesen, J. , 333 , 100 5 Kupin, S. , 177 , 178 , 197, 203, 206, 273, 937, 994, 101 0 Kaashoek, M. , 333 , 375, 987, 100 1 Kac, G. , 287 , 100 8 Lacroix, J. , 605 , 616, 617, 847 , 989, 991, Kac, L , 706 , 100 8 1010 Kac, M. , 136 , 171 , 331, 333, 368, 375, 747, Ladik, J. , 782 , 808, 98 3 782, 100 8 Lagrange, J.-L. , 222 , 101 1 Kadanoff, L. , 863 , 100 9 Laha, R,. , 410 , 101 1 Kailath, T. , 6 , 9 , 10 , 71, 143, 313, 983, 989, Lakaev, S. , 375 , 98 4 995, 1008 , 101 5 Lalesco, T., 55 , 101 1 Kaiton, N. , 216 , 100 0 Lamperti, J. , 410 , 101 1 AUTHOR INDE X 1035 Landau, H. , xiii, 69 , 105 , 101 1 Marchenko, V. , 622 , 623, 625, 630, 782 , Landkof, N. , 403 , 101 1 1014 Lanford, O. , 142 , 101 1 Marcus, B. , 605, 101 2 Langer, H. , 9 , 101 0 Margulis, G. , 616 , 100 1 Laptev, A. , 172 , 178 , 375, 937, 101 1 Markov, A. , 24 , 221, 101 4 Last, Y. , xii , 197 , 206, 286 , 287 , 563, 597, Maroni, R, 684 , 101 3 605, 606 , 624, 625, 630, 631, 638, 639, Marple, S. , 71 , 143, 101 4 647, 651, 652, 672, 707 , 723, 825, 826, Martinelli, F. , 22 , 409, 605, 100 9 845, 855 , 868, 869, 994, 1002 , 1006 , Martinez-Finkelshtein, A. , 8 , 387 , 984 , 101 4 1009, 101 1 Mate, A. , 107 , 134 , 143 , 144 , 151 , 155, 467, Lax, R, 38 , 782, 101 1 474, 493, 503, 521, 101 4 Lebowitz, J. , 351 , 992 Mathias, R. , 9 , 101 4 Ledrappier, F. , 616 , 101 1 Matsuda, H. , 581 , 100 5 Lenard, A. , 332 , 101 1 Mattila, R, 206 , 101 5 Lenz, D. , 838 , 862, 863, 992, 1011 , 1012 Matveev, V. , 782 , 807 , 987, 996, 1006 , 101 5 Le Page, E. , 616 , 101 2 Mazel, D. , 82 , 89, 101 5 Lesch, M. , 216 , 101 2 Mazenko, G. , 352 , 101 5 Lev-Ari, H. , 9 , 98 3 McCarthy, J. , 799 , 98 3 Levermore, C, 782 , 808 , 101 5 McCoy, B., 6 , 117 , 101 5 Levin, E. , 18 , 101 2 McKean, H. , 782 , 801, 807, 101 5 Levinson, N. , 6 , 70 , 216 , 101 2 McLaughlin, D. , 782 , 99 8 Levitan, B. , 216, 630, 684, 782 , 815, 992, McLaughlin, K. , 387 , 403, 544, 986 , 1014 , 999, 101 2 1015 Levy, Y., 558 , 99 4 Mehta, M. , 7 , 68 , 71, 351, 362, 101 5 Li, X. , 89 , 100 5 Mejlbo, L. , 136 , 101 5 Libkind, L. , 171 , 333, 101 2 Melik-Adamyan, F. , 9 , 101 0 Lidskii, V. , 55 , 101 2 Melman, A. , 136 , 101 5 Lieb, E. , 700 , 701, 937, 1005 , 101 2 Mersereau, R. , 143 , 99 6 Liggett, T. , 605 , 101 2 Meyer, Y. , 333 , 100 6 Lind, D. , 605 , 101 2 Mhaskar, H. , 403 , 101 5 Littlewood, J. , 55 , 100 3 Miller, R, 403 , 782, 808, 101 5 Lloyd, N. , 107 , 101 2 Milne-Thomson, L. , 520 , 101 5 Lopez, G. , 8 , 286 , 388, 389, 504 , 511 , 512, Milnor, J. , 107 , 801, 101 5 Miha, E. , 529 , 544 , 98 7 521, 534-536 , 538 , 895, 986, 987, 993, Minami, N. , 413, 101 5 1003, 1004 , 101 2 Minguez, J. , 8 , 98 7 Lothaire, M. , 603 , 605, 101 2 Miranda, R. , 741 , 763, 782, 101 5 Lowdenslager, D. , 9 , 156 , 100 4 Miura, R. , 782 , 807, 937, 998, 101 5 Lowner, K. , 799 , 101 2 Molchanov, S. , 413 , 101 5 Lubinsky, D. , 18 , 88, 107 , 189 , 403, 1012 , Moral, L. , xiii , 135 , 239, 262 , 273, 274, 975, 1013 979, 99 0 Lukashov, A. , 782 , 101 3 Moran, W., 191 , 197, 98 9 Lumer, G. , 156 , 101 3 Moreno-Balcazar, J. , 8 , 985 , 1013 , 101 4 Lyapunov, A. , 718 , 101 3 Morf, M. , 143 , 313, 1008 , 101 5 Lyons, R. , 155 , 351, 101 3 Morse, A. , 801 , 101 6 Morse, M. , 605 , 101 6 Macdonald, L , 351 , 101 3 Moser, J. , 409 , 411, 663, 672, 707 , 1006 , Mackens, W. , 136 , 101 3 1016 Magnus, A. , 189 , 101 3 Mullikin, T. , 375 , 101 6 Magnus, W. , 344 , 718 , 723, 101 3 Mumford, D. , 782 , 101 6 Makarov, K. , 216 , 375, 558, 984, 987, 100 0 Murdock, W., 136 , 100 8 Makarov, N. , 559 , 563, 994 Malamud, M. , 216 , 1012 , 101 3 Naboko, S. , xii , 178 , 829, 834, 937, 1011, Malaschonok, G. , 39 , 222 , 98 4 1016 Manakov, S. , 782 , 101 7 Nadkarni, M. : 588 , 101 6 Marcellan, F. , 8 , 24 , 89, 107 , 135 , 238, 684, Nakao, S. , 409 , 101 6 984, 985 , 987, 990 , 993, 998, 1001, Natanson, L , 206 , 101 6 1012-1014 Nazarov, F. , 925 , 937, 101 6 1036 AUTHOR INDE X Nehari, Z. , 345 , 101 6 Pinter, F. , 34 , 70 , 89, 230 , 239, 261, 1002, Nelson, E. , 690 , 101 6 1019 Nenciu, I. , 558 , 801, 807, 808, 881, 888, Pitaevskii, L. , 782 , 101 7 1009, 101 6 Plemelj, J. , 313 , 101 9 Nevai, R, xi , xiii, 11 , 18, 22, 24, 34, 56 , 70 , Poincare, H. , 512 , 520, 101 9 89, 90 , 105-107 , 134 , 143 , 144, 151, Pollack, A. , 107 , 100 3 155, 230-232 , 234 , 238, 239, 261, 262, Pollaczek, F., 178 , 101 9 273, 285 , 287, 386, 467, 474, 493, 503, Pollicott, M. , 605 , 101 9 511, 521 , 529, 550 , 558, 639, 664, 672, Poltoratski, A. , 869 , 101 9 682, 684 , 685, 824, 825, 847, 892, 893, Polya, G. , 55 , 116 , 1003 , 101 9 895, 902 , 937 , 975, 978, 995, 997, 1002 , Prüfer, H. , 672 , 101 9 1014, 1016 , 1017 , 101 9 Puig, J. , 707 , 101 9 Nevanlinna, R. , 799 , 101 7 Newell, A. , 782 , 98 3 Queffelec, M. , 602 , 605, 101 9 Newton, R. , 630 , 685, 998, 101 7 Niewiadomska-Bugaj, M. , 410 , 98 7 Rahman, M. , 89 , 99 9 Nikishin, E. , 70 , 216 , 403, 664, 672, 855, Raikov, D. , 308 , 310, 99 9 937, 985 , 101 7 Rains, E. , 7 , 98 6 Njastad, O. , 9 , 39, 69, 129 , 135 , 171, 238, Rakhmanov, E. , 9 , 11 , 121, 134, 285, 403, 239, 273 , 990, 993, 1006, 100 7 474, 503 , 975, 978, 1002 , 1019 , 102 0 Novikov, L , 333 , 101 7 Raman, S. , 375 , 102 0 Novikov, S. , 782 , 807 , 996, 101 7 Rand, D. , 863 , 101 7 Novo, S. , 216 , 100 6 Ransford, T. , 403 , 741, 102 0 Nudelman, A. , 9 , 101 7 Rao, R. , 375 , 1016 , 102 0 Nuttall, J. , 317 , 101 7 Raugi, A. , 616 , 100 3 Reed, M. , 27 , 40, 44, 45, 47, 50, 51, 55, 261, O'Connor, A. , 690 , 101 7 277, 286 , 517 , 558, 622, 623, 625, 626, Obaya, R. , 216 , 100 6 685, 691 , 718, 723, 724, 834, 102 0 Okikiolu, K. , 172 , 375, 1003 , 101 7 Reich, E. , 313 , 102 0 Okounkov, A. , 344 , 98 8 Reichel, L. , 413 , 985 Osceledec, V. , 605 , 606, 101 7 Remling, C. , xii , 563 , 673, 991, 102 0 Osher, S. , 6 , 101 7 Rezola, M. , 8 , 98 4 Osilenker, B. , 8 , 101 4 Riesz, F., 37 , 38, 40, 54 , 160 , 170 , 194 , 197 , Ostlund, S. , 863 , 101 7 1020 Ostrovsky, L , 782 , 101 4 Riesz, M. , 134 , 160 , 170 , 102 0 Oteo, J. , 344 , 101 7 Robert, D. , 172 , 375, 1011 , 102 0 Robinson, D. , 142 , 101 1 Pakula, L. , 403 , 101 7 Robinson, E. , 7 , 10 , 102 0 Pan, K. , 8 , 9 , 998 , 1017 , 101 8 Rodman, L. , 216 , 102 1 Pandit, R. , 863 , 101 7 Rogers, C. , 206 , 638, 102 1 Parter, S. , 6 , 136 , 101 8 Rogers, L. , 88 , 102 1 Pastur, L. , 409 , 605, 672, 706 , 847, 987, Rohatgi, V. , 410 , 101 1 1003, 101 8 Ros, J. , 344 , 101 7 Pearson, D. , 189 , 605, 638, 639, 647, 845, Rosenblatt, M. , 170 , 171 , 100 3 1001, 1009 , 101 8 Rosenblum, M. , 155 , 102 1 Peherstorfer, F. , 89 , 238, 239, 250, 684, Rothe, H. , 88 , 102 1 718, 719 , 729 , 735 , 741, 782, 783 , 798, Rourke, D. , 808 , 102 1 825, 829 , 888, 925, 937, 1013 , 1016 , Rovnyak, J. , 155 , 102 1 1018, 101 9 Roy, R. , 88 , 89, 98 5 Peller, V. , 346 , 101 9 Rozenblum, G. , 700 , 102 1 Perez, C, 798 , 99 0 Rudin, W. , 29 , 37, 132 , 145 , 150 , 158 , 163, Perez, T. , 8 , 101 4 164, 333, 397, 501, 660, 837, 923, 932, Perron, O. , 520 , 101 9 1021 Peyriere, J. , 197 , 101 9 Rudnick, Z. , 376 , 102 1 Phillies, G- , 352 , 101 9 Ruedemann, R. , 684 , 100 5 Pick, G. , 799 , 101 9 Ruelle, D. , 605 , 606, 102 1 Pinar, M.A. , 8 , 101 4 Ryan, R. , 333 , 100 6 Pincus, J. , 333 , 99 1 Rybalko, A. , 9 , 984 , 102 1 AUTHOR INDE X 1037 Sabatier, R, 630 , 99 1 1002, 1005 , 1009-1011 , 1016, 1020 , Safarov, Yu. , 172 , 375, 101 1 1022-1024 SafT, E. , 18 , 88, 105 , 171 , 387-389, 403, Sinai, Ya. , 707 , 99 1 512, 521 , 587, 718, 787, 986 , 988, 1007 , Sinap, A. , 135 , 102 7 1013-1015, 102 1 Singer, L , 156 , 98 5 Safronov, O. , 178 , 937, 1011 , 1021 Singh, S. , 317 , 101 7 Sakhnovich, L. , 9 , 216 , 987, 1000 , 102 1 Smirnov, V., 8 , 40 , 151, 102 4 Sansigre, G. , 89 , 101 4 Smorodinsky, M. , 588 , 102 4 Santos-Leon, J. , 135 , 102 2 Sorokin, V., 403 , 101 7 Sarason, D. , 7 , 799 , 102 2 Soshnikov, A. , 351 , 102 4 Sard, A. , 801 , 102 2 Souillard, B. , xii, 189 , 558, 855, 99 4 Sarnak, R, 351 , 376, 1008 , 102 1 Spencer, T., 285 , 286, 707 , 998, 102 3 Schatten, R. , 55 , 102 2 Spitzer, F., 313 , 376, 1022 , 102 4 Schellnhuber, H. , 863 , 101 7 Spivak, M. , 107 , 102 4 Schlag, W., 707 , 989, 100 1 Stahl, H. , 8 , 317 , 403, 718, 787, 102 4 Schmidt, R, 136 , 313, 1015, 102 2 Stanley, R. , 351 , 102 4 Schneider, A. , 216 , 100 4 Stauffer, D. , 352 , 99 1 Sch'nol, L , 286 , 102 2 Steele, J.M. , 605 , 102 4 Schur, L , 2 , 10 , 11 , 38, 39, 55, 107 , 299 , Steif, J. , 351 , 101 3 317, 351, 975, 977, 978, 998, 102 2 Stein, E. , 765 , 102 4 Schwabl, F. , 352 , 102 2 Steinbauer, R. , 89 , 238, 239, 684, 718 , 719, Schwartz, J. , 40 , 685, 99 6 729, 735 , 741, 798, 825, 829, 1018 , 101 9 Schwinger, J. , 700 , 102 2 Steinhardt, A. , 171 , 1007 Scoppola, E. , 863 , 98 7 Steklov, V. , 121 , 134, 102 4 Scott, A. , 808 , 101 5 Stieltjes, T. , 10 , 24, 251, 102 4 Segal G. , 782 , 102 2 Stollmann, R, 838 , 101 2 Segur, H. , 782 , 98 3 Stolz, G. , 550 , 605, 639, 652, 825 , 994, Seiler, E. , 55 , 102 2 1023, 102 4 Seiler, R. , 706 , 825, 98 6 Stone, M.H. , 24 , 102 4 Semencul, A. , 313 , 100 1 Strahov, E. , 685 , 98 6 Semenov, E. , 333 , 101 7 Stroock, D. , 410, 102 4 Serra, S. , 136 , 102 2 Sturm, C. , 24 , 102 4 Shahshahani, M. , 351 , 995 Suetin, R, 8 , 102 4 Shakarchi, R. , 765 , 102 4 Sukavanam, N. , 375 , 102 0 Shaw, J. , 216 , 100 4 Suto, A. , 863 , 102 4 Shen, X. , 333 , 102 7 Sylvester, J. , 39 , 222, 102 5 Shilov, G. , 308 , 310, 99 9 Sz.-Nagy, B. , 40, 102 0 Shinbrot, M. , 312 , 99 5 Szabados, J. , 107 , 102 5 Shohat, J. , 24 , 892, 1004 , 102 2 Szego, G. , xiv , 2 , 6-11, 24, 26 , 69-71, 88 , Sigal, L , 299 , 98 6 89, 105 , 109 , 116 , 121 , 124, 134 , 136 , Siggia, E. , 863 , 101 7 141, 143 , 144 , 151 , 155, 171 , 178, 321, Silbermann, B. , 6 , 9 , 142 , 313, 332, 333, 331, 332 , 376, 408, 474, 873, 880, 891, 379, 989 , 99 7 892, 975-978 , 1003 , 1008 , 102 5 Simkani, M. , 403 , 988 Simon, B. , xi-xiii, 7 , 11 , 13, 14, 16 , 17 , 20, Ta'asan, S. , 375 , 99 6 22, 24 , 25, 27, 29, 40, 44, 45, 47, Tanaka, S. , 782 , 99 3 50-52, 55 , 67, 68, 99, 104 , 105 , 107 , Tang, C. , 863 , 100 9 115, 142 , 143 , 151, 177, 178 , 184 , 189 , Taylor, M. , 847 , 102 3 197, 216 , 219 , 232 , 239, 261, 274, 277 , Taylor, S. , 206 , 638, 102 1 285-287, 293, 332, 349-351, 409, 511, Tcheremchantsev, S. , 869 , 102 5 517, 529 , 550, 558 , 559, 563, 581, 597, Teicher, H. , 410 , 99 1 600, 605 , 606, 622-626, 630 , 639, 647, Temme, N. , 90 , 102 5 651, 652 , 663, 672, 684, 685, 691, 700, Teplyaev, A. , 9 , 261, 293, 485, 550, 558, 701, 706 , 707, 718, 723, 724, 801, 807, 605, 663, 847, 855, 978, 999, 1025, 102 6 812, 815 , 825, 826, 834, 838, 845, 847, Teschl, G. , 684 , 782 , 990, 1000 , 102 6 855, 869 , 899, 903, 911, 912, 915, 918, Testard, D. , 863 , 987 920, 923 , 925, 926, 928, 931, 936-938, Theis, W. , 685 , 102 6 944, 973, 981, 986, 987, 992-994, 1000 , Thirring, W. , 808 , 102 6 1038 AUTHOR INDE X Thomas, L. , 690 , 701 , 937, 992, 100 5 Widom, H. , 6-9 , 136 , 171 , 172, 313, 332, Thouless, D. , 409 , 102 6 333, 344 , 351, 375, 379, 403, 544, 741, Thron, W. , 39 , 69, 129 , 135 , 171 , 229, 238, 987, 1026 , 102 8 239, 273 , 100 7 Wieand, K. , 351 , 102 8 Thue, A. , 605 , 102 6 Wiener, N. , 6 , 312 , 102 8 Titchmarsh, E.C. , 408 , 102 6 Wigner, E. , 684 , 812, 102 7 Toda, M. , 723 , 782, 807, 102 6 Wilf, H. , 136 , 102 8 Toeplitz, O., 37, 38, 1026 Wilson, G. , 782 , 102 2 Tomcuk, Ju., 544, 984, 1026 Wiman, A. , 824 , 102 8 Topsoe, F., 142, 313, 1026 Winkler, S. , 718 , 723, 101 3 Torrano, E., 534, 986 Witte, N. , 7 , 90, 998, 100 5 Totik, V., 8, 99, 105, 107, 134, 143, 151, Wittwer, R , 707 , 99 8 155, 184, 189, 386, 403, 467, 474, 493, Woerdeman, H. , 38 , 99 9 503, 587, 682, 685, 718, 741, 787, 893, Wojtaszczyk, R , 333 , 102 8 895, 902, 1014, 1017, 1021, 1023, 1024, Wolff, T. , 550 , 558 , 847, 102 3 1026 Wu, J. , 107 , 101 0 Tracy, C, 351, 1026 Yakovlev, S. , 834 , 101 6 Trench, W., 136, 313, 333, 1026 Yandell, B. , 580 , 102 8 Trubowitz, E., 741, 816, 998, 1026 Youla, D. , 8 , 216 , 102 8 Tsang, T., 352, 1026 Yuditskii, R , 925 , 937, 1016 , 101 9 Tsekanovskii, E. , 216 , 100 0 Tsuji, M. , 403 , 102 6 Zaharov, V. , 782 , 937, 1017 , 102 9 Turan, R , 22 , 97, 107 , 403, 997, 102 6 Zamfirescu, T. , 835 , 102 9 Zeilberger, D. , 222 , 984, 102 9 Ushiroya, N. , xii, 598 , 606, 855, 101 0 Zelditch, S. , 172 , 1006 , 102 9 Uvarov, V. , 684 , 685, 102 7 Zhang, J. , 11 , 90, 105 , 106 , 978, 99 7 Zhou, X. , 403 , 544, 986 , 99 3 Van Assche , W., 8 , 89, 135 , 189 , 261, 262, Zinchenko, M. , 816 , 100 0 273, 993 , 1002 , 1012 , 1013 , 102 7 Zlatos, A. , 177 , 197 , 926, 928, 936, 937 , van Moerbeke , R , 7 , 723, 741, 747, 782 , 981, 1023 , 1024 , 102 9 801, 807 , 983, 1008 , 1015 , 1016 , 102 7 Zygmund, A. , 197 , 102 9 van Schagen , F. , 333 , 375, 100 1 Vaninsky, K. , 808 , 102 7 Velazquez, L. , xiii , 135 , 239, 262, 273, 274, 975, 979 , 99 0 Verblunsky, S. , xiii , 7 , 10 , 11 , 70, 88, 106 , 107, 141 , 217, 221, 222, 238 , 975, 977, 978, 102 7 Vieira, A. , 143 , 313, 995, 1008 , 101 5 Vigil, L. , 97 , 107 , 98 4 Volberg, A. , 925 , 937, 101 6 von Neumann , J. , 54 , 55, 684, 812 , 1022 , 1027 Voss, H. , 136 , 1013 , 102 7 Waadeland, H. , 135 , 171, 1007, 102 7 Wall, H.S. , 39 , 69, 102 7 Walsh, J. , 24 , 100 4 Walter, G. , 333 , 102 7 Wang, W.-M. , 413 , 102 7 Wegner, F. , 557 , 102 7 Weidl, T. , 701 , 937, 102 7 Weidmann, J. , 639 , 817, 824, 102 7 Weiss, B., 605, 100 9 Wendroff, B. , 15 , 24, 102 7 Wermer, J. , 156 , 402, 403, 102 7 Weyl, H. , 55 , 351, 1027, 102 8 Subject Inde x Abel condition , 76 3 bound state , 62 3 Abel map, 76 2 boundary condition , 222 , 259, 26 9 Abel's formula , 66 6 boundary value , 2 9 Abel's theorem , 76 2 bounded Variation , 667 , 81 7 abelian period , 76 0 Boyd's theorem , 16 3 absolutely continuou s measure , 4 3 AKV lemma , 21 7 canonical decomposition , 4 6 Aleksandrov measure , 35 , 54, 222 , 234, 238, Cantor set , 19 9 269, 545 , 551, 552, 773, 802 capacity, 40 2 Alfaro-Vigil theorem , 9 7 Caratheodory function , 3 , 25, 28, 36, 225, analytic vector , 68 6 382, 552 , 790 , 79 2 antilinear Operator , 4 0 m-Caratheodory function , 29 4 anti-unitary Operator , 4 0 minimal Caratheodor y function , 76 7 approximate densit y o f zeros, 39 1 Caratheodory-Toeplitz theorem , 26 , 38, 21 7 Aptekarev's theorem , 73 5 Carmona's criterion , 620 , 83 9 Aronszajn-Donoghue theory , 545-55 0 Cauchy inequality , 11 5 associated polynomial , 24 5 Cayley transform , 42 , 69 5 CD kernel , 12 4 Baire categor y theorem , 83 4 Cesäro approximation , 32 8 Baker-Campbell-Hausdorff formula , 34 4 Cesäro average , 110 , 40 7 balayage, 40 4 character, 34 9 Banach algebra , 30 8 Chebyshev polynomial s o f the fourt h kind , band, 71 0 888 Baxter's lemma , 30 4 Chebyshev polynomial s o f the secon d kind , Baxter's theorem , 4 , 6 , 33 , 31 3 13 Bernstein inequality , 12 1 Chebyshev polynomial s o f the thir d kind , Bernstein-Szegö approximation , 95 , 122 , 888 143, 148 , 225, 468, 47 8 Chebyshev's theorem , 81 3 Bernstein-Szegö measure , 111 , 320 Christoffel function , 16 , 117 , 124 , 169 , 49 3 Bernstein-Szegö polynomial , 72 , 8 8 Christoffel-Bargmann perturbations , 673 , Beurling algebra , 307 , 32 1 680 Beurling weight , 306 , 311, 312, 31 4 Christoffel-Darboux formula , 18 , 60, 124 , Birman-Schwinger bound , 69 1 224, 403, 493, 51 8 Birman-Schwinger principle , 69 0 circuit theory , 6 Blaschke factor , 92 1 CMV basis , 263 , 291, 882 Blaschke product , 25 , 30, 36, 796 , 94 4 CMV factorization , 88 2 Blatt-Saff-Simkani lemma , 39 5 CMV matrix , 287 , 293, 629, 640, 685, 800, Blumenthal-Weyl condition , 92 6 808, 88 2 Bochner's theorem , 3 8 extended CM V matrix , 294 , 589 , 704 , Boole's equality , 86 7 719 Borel transform , 1 2 CMV representation , 264 , 27 4 Borel-Cantelli lemma , 40 6 alternate CM V representation , 26 4 Borg's secon d theorem , 81 3 coefficient Stripping , 245 , 25 9 Borodin-Okounkov formula , 336 , 34 1 Combes-Thomas theory , 69 0 1039 1040 SUBJECT INDE X commutant liftin g theorems , 7 Favard's theorem , 2 , 14 , 25 1 compact Operator , 4 5 Fejer approximation , 20 3 concave, 13 8 Fejer kernel , 20 6 concave function , 11 1 Fejer's theorem , 103 , 328 continued fraction , 20 , 69, 229 , 235, 71 8 Fejer-Riesz theorem , 26 , 38, 94, 13 5 convolution, 60 7 Fenchel's theorem , 14 2 Cotes number , 1 7 Feshbach projection , 29 9 Coulomb energy , 35 5 Fibonacci sequence , 602 , 85 5 Coulomb gas , 35 2 Fibonacci subshift , 85 5 Coulomb ga s representation, 6 7 filtering theory , 6 cyclic vector , 4 2 finite Jacob i matrix , 2 1 finite rank , 29 5 Damanik-Killip theorem , 89 6 Floquet theory , 71 9 Damanik-Simon theorem , 91 5 free Jacob i matrix , 1 3 degree theory, 9 8 Freudian paralle l universe , 132 , 143 , 15 0 Deift-Killip theorem , 33 8 FSW duality , 35 0 Deift-Simon theorem , 66 1 functional calculus , 4 4 Denisov an d Kupin' s Workshop , 19 7 Furstenberg condition , 607 , 61 0 Denisov-Rakhmanov theorem , 89 2 Furstenberg's theorem , 607 , 61 7 dense G$, 54 9 dense poin t spectrum , 82 9 Galois dual, 78 9 density o f states, 22 , 591, 701 Galois theorem, 793 , 794 density o f zeros, 391, 402, 404, 65 6 gap, 71 0 derived set , 5 , 4 3 gap closure , 72 4 determinant, 4 9 Gauss-Jacobi quadrature , 129 , 13 0 determinate, 1 4 Gauss-Jacobi quadratur e formula , 17 , 2 1 Devinatz's formula , 328 , 339, 345, 358, 36 3 Gaussian measure , 7 7 DHK Formula , 37 1 Gaussian rando m variable , 34 7 direct Geronimu s relations , 87 5 Gel'fand spectrum , 30 8 Dirichlet algebra , 15 6 generic singula r continuou s spectrum , 83 4 Dirichlet approximation , 20 3 genus, 75 5 Dirichlet data , 731 , 742 geophysical scattering , 7 Dirichlet dat a torus , 731 , 767, 78 4 Geronimo-Case equations , 90 5 discrete spectrum , 4 3 Geronimus polynomials , 83 , 87, 89, 716 , discriminant, 710 , 715 , 74 8 728 Dodgson's equality , 21 9 Geronimus relations , 875 , 877, 88 4 doubly substochastic , 4 7 direct Geronimu s relations , 87 5 dynamical System , 60 0 inverse Geronimu s relations , 87 7 Geronimus' theorem , 3 , 179 , 219 , 226, 229, eigenfunction expansion , 62 4 247, 29 8 eigenvalue, 16 0 Geronimus-Wendroff theorem , 1 5 elliptic, 565 , 56 8 GGT matrix , 25 2 energy, 39 3 GGT representation , 25 2 entropy, 136 , 60 8 füll GG T representation , 25 6 semicontinuity, 13 8 Gl approximation , 32 2 equilibrium measure , 402 , 503, 710, 71 5 Gl measure , 32 2 equilibrium potential , 710 , 71 5 Gibbs principle , 14 2 essential spectrum , 5 , 24 8 Gilbert-Pearson theorem , 63 9 essential support , 4 3 Golinskii's formula , 22 6 Euler's formula , 33 1 Golinskii-Ibragimov theorem , 32 1 Euler's theorem , 79 0 Gordon-del Rio-Makarov-Simo n theorem , Euler-Wallis formulae , 39 , 6 9 558 exact dimension , 19 9 Green's function , 41 , 640, 81 0 exact leadin g asymptotics , 9 1 group representatio n theory , 34 9 exponential decay , 381 , 912 extended CM V matrix , 294 , 589 , 71 9 Hamiltonian flow, 80 3 extreme point , 16 4 Hankel matrix , 11 , 15, 333 Hankel Operator , 333 , 334, 336 , 34 4 F. an d M . Ries z theorem, 16 0 Hardy-Littlewood maxima l inequality , 83 7 SUBJECT INDE X 1041 harmonic measure , 73 4 Khrushchev's formula , 287 , 298, 477, 479 , harmonic measur e criterion , 73 4 521, 65 6 harmonic midpoints , 74 0 Khrushchev's Workshop , 18 9 Hausdorff dimension , 188 , 199 , 635, 649, Killip-Simon theorem , 828 , 92 5 863 Kolmogorov's theorem , 55 6 Hausdorff measure , 19 9 Kooman's theorem , 818 , 82 3 Hausdorff-Young inequality , 32 3 Kotani theory , 65 2 Heine's formula , 15 , 65 Kotani's firs t theorem , 652 , 655, 70 2 Helton-Howe theorem , 340 , 34 2 Kotani's fourt h theorem , 653 , 660 Herglotz function , 1 2 Kotani's secon d theorem , 653 , 65 9 Herglotz representation , 28 , 3 8 Kotani's thir d theorem , 653 , 66 0 Hermite polynomial , 1 3 Krein algebra , 34 4 Hessenberg matrix , 252 , 25 4 Krein system , 7 , 9 high barriers , 86 3 Krein's theorem , 14 1 Hilbert-Schmidt Operator , 52 , 33 9 KW pair , 23 9 Hilbert-Schmidt theorem , 4 6 Holder continuous , 32 9 Last-Simon criterion , 64 1 Holder's inequality , 5 2 Laurent polynomial , 25 , 87 3 hyperbolic, 565 , 56 8 Legendre polynomial , 1 3 hyperbolic geometry , 58 0 Legendre transform , 14 2 hyperboloid, 56 8 Levinson algorithm , 6 7 hyperelliptic Rieman n surface , 73 5 Lidskii's theorem , 51 , 55 Lieb-Thirring condition , 92 6 Ibragimov's theorem , 321 , 342, 36 8 linear predictio n theory , 16 7 inner function , 3 6 CM. factorization , 26 5 inserted mas s point , 7 2 logmodular algebra , 15 6 invariant measure , 573 , 607 Lopez condition , 504 , 50 6 inverse Geronimu s relations , 87 7 Lopez ^-fol d condition , 51 0 inverse Peherstorfer' s formula , 24 7 lower semicontinuous , 393 , 54 9 inverse Schu r iterate , 47 6 lower triangulär, 30 2 inverse Szeg ö recursion, 5 9 Löwner order , 4 5 Ishii-Pastur theorem , 58 4 Lyapunov exponent , 583 , 585, 593, 701, 711 isospectral, 73 0 isospectral flow, 735 , 80 1 m-Caratheodory function , 29 4 isospectral torus , 79 9 m-function, 12 , 1 9 isospectral toru s theorem , 73 2 m-Schur function , 29 4 Mäte-Nevai condition , 48 5 J-invariance, 58 , 56 4 mass point, 4 3 J-unitarity, 5 8 inserted mas s point, 7 2 Jacobi inversio n theorem , 76 5 matrix-valued measure , 206 , 21 2 Jacobi matrix , 13 , 251, 884, 899, 912, 93 8 matrix-valued polynomial , 8 finite Jacob i matrix , 2 1 meromorphic function , 75 6 free Jacob i matrix , 1 3 meromorphic Herglotz , 92 0 Jacobi parameters , 871 , 875, 904, 91 2 MH function , 92 0 Jacobi's relatio n o f minors, 22 0 Mhaskar-Saff theorem , 392 , 41 2 Jensen's inequality , 119 , 143 , 154 , 21 3 min-max, 4 4 Jentzsch-Szegö theorem , 40 9 minimal, 60 0 Jitomirskaya-Last inequalities , 631 , 633 minimal Caratheodor y function , 76 7 Jost asymptotics , 93 8 minimum problem , 120 , 16 5 Jost expansion , 62 3 mixed C D formula , 22 4 Jost function , 623 , 628, 904, 909, 91 2 MNT theory , 49 3 Jost Solution , 623 , 627, 904, 90 9 modulus, 239 , 73 8 Jost-Kohn perturbations , 67 3 moment, 1 1 monic orthogonal polynomial , 5 5 Kato similarity , 82 5 multiplicity 2 , 70 5 Kato-Birman theorem , 5 3 mutually singular , 54 6 for OPUC , 27 7 Kato-Gilbert theorem , 70 6 Naboko's Workshop , 82 9 Khrushchev condition , 48 5 Nehari's criterion , 34 5 1042 SUBJECT INDE X Nehari's theorem , 33 4 random matri x theory , 6 Nevai's conjecture , 17 8 rank on e matrix , 29 4 Nevai-Totik radius , 38 3 rank on e perturbation , 5 3 Nevai-Totik theorem , 383 , 912 rank tw o perturbation, 29 3 Nevanlinna function , 12 , 93 1 ratio asymptotics , 91 , 504, 53 2 nontrivial measure , 1 rearrangement inequalities , 4 7 normal Operator , 4 0 reflection, 57 4 nowhere dense , 55 9 regulär point , 98 , 79 9 regulär Solution , 62 2 OPRL, 1 1 regulär value , 98 , 79 9 orbit, 60 0 relative entropy , 136 , 16 9 order, 75 6 relative Szeg ö function , 18 0 orthogonal moni c polynomial , 1 2 renormalized determinant , 52 , 55, 27 2 orthogonal rationa l function , 8 reproducing kernel , 16 , 120 , 49 3 orthonormal polynomial , 12 , 5 5 resolvent, 40 , 28 7 outer function , 3 7 restricted densit y o f zeros, 39 1 reversed polynomial , 2 P2 sum rule , 92 5 Riemann surface , 75 3 Pade approximant , 229 , 23 8 hyperelliptic Rieman n surface , 75 7 parabolic, 565 , 56 8 Riemann-Hilbert problems , 33 2 paraorthogonal polynomial , 129 , 130 , 40 7 Riesz product, 189 , 19 1 Peherstorfer's formula , 24 6 right limit , 60 0 inverse Peherstorfer' s formula , 24 7 right limi t point , 64 4 Peherstorfer-Steinbauer theorem , 22 8 Rogers-Szegö polynomial , 8 7 Peierls-Bogoliubov inequality , 21 6 Rogers-Szegö polynomial , 77 , 82, 8 8 period lattice , 76 1 root asymptotics , 9 1 period vector , 76 0 root free , 75 7 Pick function , 1 2 rotation number , 410 , 41 1 Pick interpolation , 79 9 Ruelle-Osceledec theorem , 595 , 606, 84 5 Pick's theorem, 79 9 Pinter-Nevai formula , 22 9 5-matrix, 344 , 625, 62 6 Poincare indices , 51 2 Sard's theorem , 79 9 Poincare polynomial , 51 2 scattering theory , 61 7 Poincare's theorem , 51 2 Schur algorithm , 30 , 39, 179 , 297, 79 7 Poincare-Perron theorem , 51 6 Schur approximant , 31 , 35, 492 Poisson bracket , 801 , 802 Schur basis , 47 , 5 1 Poisson distribution , 41 3 Poisson kernel , 27 , 118 , 151, 404, 411, 637 Schur function , 3 , 6 , 25 , 30, 31, 36, 163, Poisson representation , 2 7 164, 169 , 235, 248, 297, 298, 314, 475, polar decomposition , 4 6 492, 521 , 659, 79 0 polynomial ratios , 46 7 m-Schur function , 29 4 positive Operator , 4 5 Schur iterate , 30 , 180 , 48 1 Potential, 39 3 inverse Schu r iterate , 47 6 Potential theory , 393-39 6 Schur parameter , 3 , 6, 30 , 219 , 52 1 Prüfer variable , 83 9 Schur's recurrenc e relation , 3 1 Prüfer variables , 66 4 Schur-Lalesco-Weyl inequalities , 4 8 pure point , 43 , 54 7 second kin d polynomial , 18 , 222, 22 7 second unitarit y condition , 35 0 quacks-like-a-discriminant theorem , 73 3 selfadjoint Operator , 4 0 quadratic irrationality , 789 , 79 2 sieved polynomial , 84 , 8 9 quasi-Szegö condition , 92 6 Simon-Wolff criterion , 55 1 quasi-unitary, 81 8 singular continuou s measure , 4 3 singular inne r function , 3 6 r-growing, 19 0 singular measure , 43, 15 2 Rakhmanov condition , 485 , 48 8 singular spectrum , 70 6 Rakhmanov's lemma , 260 , 276, 643, 86 3 Sobolev space , 32 9 Rakhmanov's theorem , 5 , 471, 481, 530, spectral averaging , 55 1 535, 828 , 89 2 spectral invariant , 79 9 random matri x product , 60 6 spectral measure , 42 , 62 4 SUBJECT INDE X 1043 spectral theorem , 4 2 transfer matrix , 18 , 224, 58 1 spectral theory , 7 trial function , 280 , 28 4 spectrum, 4 0 triple produc t formula , 7 9 stationary stochasti c process , 6 Turän measure , 98 , 10 7 Steklov conjecture , 12 1 two-point function , 36 2 step-by-step su m rule , 92 4 Stieltjes momen t problem , 14 2 U(l,l), 56 4 Stieltjes transform , 1 2 unique representin g measure , 15 5 Stone's theorem , 68 6 uniquely ergodic , 60 0 strictly ergodic , 60 0 unitary Operator , 4 0 strong Operato r convergence , 4 1 unitary Operators , 7 strong Szeg ö theorem, 4 , 321, 348, 368, 37 5 upper semicontinuity , 60 8 strongly continuou s unitar y group , 68 6 upper semicontinuous , 13 8 strongly irreducible , 60 9 upper triangulär , 30 2 strongly irreducibl e measure , 60 9 Vandermonde determinant , 68, 115 , 35 4 Sturm compariso n theorem , 2 4 variational principle , 13 7 Sturm oscillatio n theorem , 2 2 varying measure , 53 1 Sturmian sequence , 60 3 Verblunsky coefficients , 2 , 56, 67, 210 , 871, SU(1, 1), 56 4 875 subadditive ergodi c theorem, 59 2 almost periodi c Verblunsk y coefficients , subordinate Solution , 63 9 589, 70 6 subshift, 60 1 minimal recurren t Verblunsk y Substitution sequence , 60 3 coefficients, 58 9 sup-norm algebra , 15 5 periodic Verblunsk y coefficients , 83 , 285, symbol, 33 4 589, 71 8 symbolic dynamics , 60 1 random Verblunsk y coefficients , 552 , 588, symplectic structure , 80 1 845 Szegö asymptotics, 144 , 891, 943 decaying rando m Verblunsk y Szegö dass, 82 9 coefficients, 84 7 Szegö condition, 143 , 147, 193 , 256, 272, sparse Verblunsk y coefficients , 83 8 314, 319 , 382, 629, 89 0 stochastic Verblunsk y coefficients , 588 , Szegö differenc e equation , 5 6 590, 593 , 701 Szegö function, 109 , 144 , 169 , 173 , 225, Verblunsky formula , 32 , 6 0 272, 314 , 320, 38 2 Verblunsky's theorem , 2 , 97, 218, 258, 26 8 Szegö mapping, 871 , 912 Verblunsky, Samuel , 22 1 Szegö recurrence, 2 Vitali convergenc e theorem , 38 5 Szegö recursion, 56 , 126 , 210, 218, 316, 477, 488, 90 4 Wall polynomial , 33 , 239, 748, 80 3 Szego's theorem, 4 , 109 , 136 , 141 , 154, 155 , wave Operator , 62 5 158, 163 , 169 , 180 , 187 , 212, 356, 540 , weak asymptoti c measure , 407 , 52 3 828, 890 , 930, 93 7 weak Operato r convergence , 4 1 weakly reducible , 60 9 Taylor coefficient , 3 1 Weidmann's theorem , 81 7 Thouless formula , 585 , 655, 701, 715 Wendroff's theorem , 9 3 three-term recurrence , 6 0 Weyl m-function, 23 1 three-term recurrenc e relation , 1 2 Weyl circle , 23 1 Thue-Morse sequence , 60 2 Weyl Integratio n formula , 6 8 Toeplitz determinant , 4 , 6 , 26 , 109 , 319, Weyl sequence , 70 3 352 Weyl Solution , 288 , 93 8 Toeplitz matrix , 6 , 26 , 135 , 168, 33 3 Weyl's theorem, 5 3 Toeplitz Operator , 9 , 302 , 313, 33 4 for OPUC , 27 7 Totik's Workshop , 18 4 WGN disk , 23 1 trace, 4 9 Widom's formula , 337 , 34 1 trace dass , 4 9 Widom's lemma , 39 7 trace dass Operator , 33 9 Widom's zer o theorem, 39 7 trace ideal , 5 1 Wiener algebra , 30 9 trace map , 85 6 Wiener Tauberia n theorem , 30 9 trace norm , 27 4 Wiener's theorem , 83 6 1044 SUBJECT INDE X 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, 24 , 56 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 e 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 Schrödinge 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 , Coemcien t region s fo r Schlich t funetions , 195 0 34 J . L . Walsh , Th e locatio n o f critical point s o f analytic an d harmoni c funetions , 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 . Radö , Lengt h an d area , 194 8 29 A . Weil , Foundation s o f algebraic 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 . Szegö , 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 funetion 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 funetions , 193 3 15 M . H . Stone , Linea r transformation s i n Hubert spac e an d thei r application s t o analysis , 1932 TITLES I N THIS 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 , Functional 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 . Wilczyriski , Projectiv e differentia l geometry , 191 0 1 H . S . White , Linea r System s o f curves o n algebrai c 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 n; see the remark followin g Theorem 1.7.1 2 Zn(d\i) zero s o f $n; se e the remark followin g Theore m 1.7.1 2 Z\^{d\i) limi t point s o f zeros o f $n; se e the remark followin g Theore m 1.7.1 2 Zsh(d/ji) strictl y limit s o f zeros o f 3> n; see 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; see (13.8.22 ) Z(J I J^) step-by-ste p entrop y i n C0 sum rule; see (13.8.24 ) Z integer s {0 , ±1, ±2,... } Z+ nonnegativ e integer s {0,1,2,... } Nonalphabet ic \A\ absolut e valu e o f an Operator; se e (1.4.28) (*;