Orthogonal Polynomials on the Unit Circle
by Barry Simon (Caltech)

 

Table of Contents
Part 1: Classical Theory
 
Preface to Part 1     xi
Notation     xvii
 

Chapter 1: The Basics
  1.1 Introduction   1
  1.2 Orthogonal Polynomials on the Real Line   11
  1.3 Caratheodory and Schur Functions   25
  1.4 An Introduction to Operator and Spectral Theory   40
  1.5 Verblunsky Coefficients and the Szego Recurrence   55
  1.6 Examples of OPUC   71
  1.7 Zeros and the First Proof of Verblunsky's Theorem   90
       
Chapter 2: Szego's Theorem
  2.1 Toeplitz Determinants and Verblunsky Coefficients   109
  2.2 Extremal Properties, the Christoffel Functions, and the
Christoffel-Darboux Formula 		 
117
  2.3 Entropy Semicontinuity and the First Proof of Szego’s Theorem   136
  2.4 The Szego Function   143
  2.5 Szego's Theorem Using the Poisson Kernel   151
  2.6 Khrushchev’s Proof of Szego’s Theorem   156
  2.7 Consequences of Szego’s Theorem   159
  2.8 A Higher-Order Szego Theorem   172
  2.9 The Relative Szego Function   178
  2.10 Totik’s Workshop   184
  2.11 Riesz Products and Khrushchev’s Workshop   189
  2.12 The Workshop of Denisov and Kupin   197
  2.13 Matrix-Valued Measures   206
         
Chapter 3: Tools for Geronimus' Theorem
  3.1 Verblunsky’s Viewpoint: Proofs of Verblunsky’s and Geronimus’ Theorems   217
  3.2 Second Kind Polynomials   222
  3.3 KW Pairs   239
  3.4 Coefficient Stripping and Associated Polynomials   245
         
Chapter 4: Matrix Representations
  4.1 The GGT Representation   251
  4.2 The CMV Representation   262
  4.3 Spectral Consequences of the CMV Representation   274
  4.4 The Resolvent of the CMV Matrix   287
  4.5 Rank Two Perturbations and Decoupling of CMV Matrices   293
         
Chapter 5: Baxter's Theorem
  5.1 Wiener-Hopf Factorization and the Inverses of Finite Toeplitz Matrices   301
  5.2 Baxter's Proof   313
         
Chapter 6: The Strong Szego Theorem
  6.1 The Ibragimov and Golinskii-Ibragimov Theorems   319
  6.2 The Borodin-Okounkov Formula   333
  6.3 Representations of U(n) and the Bump-Diaconis Proof   346
  6.4 Toeplitz Determinants as the Statistical Mechanics of Coulomb Gases
and Johansson’s Proof		  
352
  6.5 The Combinatorial Approach and Kac’s Proof   368
  6.5 A Second Look at Ibragimov’s Theorem   376
         
Chapter 7: Verblunsky Coefficients With Rapid Decay
  7.1 The Rate of Exponential Decay and a Theorem of Nevai-Totik   381
  7.2 Detailed Asymptotics of the Verblunsky Coefficients   387
         
Chapter 8: The Density of Zeros
  8.1 The Density of Zeros Measure via Potential Theory   391
  8.2 The Density of Zeros Measure via the CMV Matrix   403
  8.3 Rotation Numbers   410
  8.4 A Gallery of Zeros   412
         
Bibliography   425
Author Index   457
Subject Index   463
         
         
Part 2: Spectral Theory
        xi
Preface to Part 2   xiii
Notation    
         
Chapter 9: Rakhmanov’s Theorem and Related Issues    
  9.1 Rakhmanov’s Theorem via Polynomial Ratios   467
  9.2 Khrushchev’s Proof of Rakhmanov’s Theorem   475
  9.3 Further Aspects of Khrushchev’s Theory   485
  9.4 Introduction to MNT Theory   493
  9.5 Ratio Asymptotics   503
  9.6 Poincare’s Theorem and Ratio Asymptotics   512
  9.7 Weak Asymptotic Measures   521
  9.8 Ratio Asymptotics for Varying Measures   530
  9.9 Rakhmanov’s Theorem on an Arc   535
  9.10 Weak Limits and Relative Szego Asymptotics   538
         
Chapter 10: Techniques of Spectral Analysis    
  10.1 Aronszajn-Donoghue Theory   545
  10.2 Spectral Averaging and the Simon-Wolff Criterion   551
  10.3 The Gordon-del Rio-Makarov-Simon Theorem   558
  10.4 The Group U(1;1)   564
  10.5 Lyapunov Exponents and the Growth of Norms in U(1;1)   581
  10.5A Appendix: Subshifts   600
  10.6 Furstenberg’s Theorem and Random Matrix Products From U(1;1)   606
  10.7 The Transfer Matrix Approach to L1  Verblunsky Coefficients   617
  10.8 The Jitomirskaya-Last Inequalities     631
  10.9 Criteria for A.C. Spectrum   639
  10.10 Dependence on the Tail   648
  10.11 Kotani Theory   652
  10.12 Prufer Variables   664
  10.13 Modifying the Measure: Inserting Eigenvalues and Rational Function Multiplication         673
  10.14 Decay of CMV Resolvents and Eigenfunctions   685
  10.15 Counting Eigenvalues in Gaps: The Birman-Schwinger Principle   690
  10.16 Stochastic Verblunsky Coefficients   701
         
Chapter 11: Periodic Verblunsky Coefficients    
  11.1 The Discriminant   710
  11.2 Floquet Theory   719
  11.3 Calculation of the Weight   724
  11.4 An Overview of the Inverse Spectral Problem   730
  11.5 The Orthogonal Polynomials Associated to Dirichlet Data   742
  11.6 Wall Polynomials and the Determination of Discriminants   748
  11.7 Abel’s Theorem and the Inverse Spectral Problem   753
  11.8 Almost Periodic Isospectral Tori   783
  11.9 Quadratic Irrationalities   788
  11.10 Independence of Spectral Invariants and Isospectral Tori   799
  11.11 Isospectral Flows   801
  11.12 Bounds on the Green’s Function   808
  11.13 Genericity Results   811
  11.14 Consequences of Many Closed Gaps   812
         
Chapter 12: Spectral Analysis of Specific Classes of Verblunsky Coefficients    
  12.1 Perturbations of Bounded Variation   817
  12.2 Perturbations of Periodic Verblunsky Coefficients   826
  12.3 Naboko’s Workshop: Dense Point Spectrum in the Szego Class    829
  12.4 Generic Singular Continuous Spectrum   834
  12.5 Sparse Verblunsky Coefficients   838
  12.6 Random Verblunsky Coefficients   845
  12.7 Decaying Random Verblunsky Coefficients   847
  12.8 Subshifts   855
  12.9 High Barriers   863
         
Chapter 13: The Connection to Jacobi Matrices    
  13.1 The Szego Mapping and Geronimus Relations   871
  13.2 CMV Matrices and the Geronimus Relations   881
  13.3 Szego’s Theorem for OPRL: A First Look   889
  13.4 The Denisov-Rakhmanov Theorem   892
  13.5 The Damanik-Killip Theorem   896
  13.6 The Geronimo-Case Equations   903
  13.7 Jacobi Matrices With Exponentially Decaying Coefficients   912
  13.8 The P2 Sum Rule and Applications   920
  13.9 Szego’s Theorem for OPRL: A Third Look   937
         
Appendix A - Reader's Guide: Topics and Formulae

A.1

 What's Done Where    
  A Schur functions   945
  B Toeplitz matrices and determinants   945
  C Szego’s theorem   946
  D Aleksandrov families   946
  E Zeros of OPUC   946
  F Density of zeros   946
  G CMV matrices   947
  H Periodic Verblunsky coefficients   947
  I Stochastic Verblunsky coefficients   947
  J Transfer matrices   948
  K Asymptotics of orthogonal polynomials   948

A.2

 Formulae    
  A Basic objects   948
  B Recursion   950
  C Bernstein-Szego approximation   955
  D Additional OP formulae   956
  E Additional Wall polynomial formulae   956
  F Matrix representations   956
  G Aleksandrov families   959
  H Rotation of measure   960
  I Sieved polynomials   960
  J Toeplitz determinants (see also K)   961
  K Szego’s theory   961
  L Additional transfer matrix formulae   964
  M Periodic Verblunsky coefficients   966
  N Connection to Jacobi matrices   967
         
Appendix B - Perspectives
  B.1 OPRL vs. OPUC   971
  B.2 OPUC Analogs of the m-function   973
         
Appendix C - Twelve Great Papers   975
         
Appendix D - Conjectures and Open Questions
  D.1 Related to Extending Szego’s Theorem   981
  D.2 Related to Periodic Verblunsky Coefficients   981
  D.3 Spectral Theory Conjectures   982
         
Bibliiography   983
Author Index   1031
Subject Index   1039
         

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