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This book presents a comprehensive overview of the sum rule approach to spectral analysis of orthogonal polynomials, which derives from Gábor Szego's classic 1915 theorem and its 1920 extension. Barry Simon emphasizes necessary and sufficient conditions, and provides mathematical background that until now has been available only in journals. Topics include background from the theory of meromorphic functions on hyperelliptic surfaces and the study of covering maps of the Riemann sphere with a finite number of slits removed. This allows for the first book-length treatment of orthogonal polynomials for measures supported on a finite number of intervals on the real line.
In addition to the Szego and Killip-Simon theorems for orthogonal polynomials on the unit circle (OPUC) and orthogonal polynomials on the real line (OPRL), Simon covers Toda lattices, the moment problem, and Jacobi operators on the Bethe lattice. Recent work on applications of universality of the CD kernel to obtain detailed asymptotics on the fine structure of the zeros is also included. The book places special emphasis on OPRL, which makes it the essential companion volume to the author's earlier books on OPUC.
Preface ix
Chapter 1. Gems of Spectral Theory 1
1.1 What Is Spectral Theory? 1
1.2 OPRL as a Solution of an Inverse Problem 4
1.3 Favard's Theorem, the Spectral Theorem, and the Direct Problem for OPRL 11
1.4 Gems of Spectral Theory 18
1.5 Sum Rules and the Plancherel Theorem 20
1.6 Pólya's Conjecture and Szego's Theorem 22
1.7 OPUC and Szego's Restatement 24
1.8 Verblunsky's Form of Szego's Theorem 26
1.9 Back to OPRL: Szego Mapping and the Shohat-Nevai Theorem 30
1.10 The Killip-Simon Theorem 37
1.11 Perturbations of the Periodic Case 39
1.12 Other Gems in the Spectral Theory of OPUC 41
Chapter 2. Szego's Theorem 43
2.1 Statement and Strategy 44
2.2 The Szego Integral as an Entropy 48
2.3 Carathéodory, Herglotz, and Schur Functions 52
2.4 Weyl Solutions 66
2.5 Coefficient Stripping, Geronimus' and Verblunsky's Theorems, and Continued Fractions 74
2.6 The Relative Szego Function and the Step-by-Step Sum Rule 80
2.7 The Proof of Szego's Theorem 84
2.8 A Higher-Order Szego Theorem 86
2.9 The Szego Function and Szego Asymptotics 91
2.10 Asymptotics for Weyl Solutions 97
2.11 Additional Aspects of Szego's Theorem 98
2.12 The Variational Approach to Szego's Theorem 103
2.13 Another Approach to Szego Asymptotics 108
2.14 Paraorthogonal Polynomials and Their Zeros 113
2.15 Asymptotics of the CD Kernel: Weak Limits 118
2.16 Asymptotics of the CD Kernel: Continuous Weights 123
2.17 Asymptotics of the CD Kernel: Locally Szego Weights 132
Chapter 3. The Killip-Simon Theorem: Szego for OPRL 143
3.1 Statement and Strategy 143
3.2 Weyl Solutions and Coefficient Stripping 144
3.3 Meromorphic Herglotz Functions 151
3.4 Step-by-Step Sum Rules for OPRL 158
3.5 The P2 Sum Rule and the Killip-Simon Theorem 163
3.6 An Extended Shohat-Nevai Theorem 167
3.7 Szego Asymptotics for OPRL 173
3.8 The Moment Problem: An Aside 183
3.9 The Krein Density Theorem and Indeterminate Moment Problems 203
3.10 The Nevai Class and Nevai Delta Convergence Theorem 207
3.11 Asymptotics of the CD Kernel: OPRL on [?2, 2] 213
3.12 Asymptotics of the CD Kernel: Lubinsky's Second Approach 222
Chapter 4. Sum Rules and Consequences for Matrix Orthogonal Polynomials 228
4.1 Introduction 228
4.2 Basics of MOPRL 229
4.3 Coefficient Stripping 234
4.4 Step-by-Step Sum Rules of MOPRL 239
4.5 A Shohat-Nevai Theorem for MOPRL 244
4.6 A Killip-Simon Theorem for MOPRL 246
Chapter 5. Periodic OPRL 250
5.1 Overview 250
5.2 m-Functions and Quadratic Irrationalities 253
5.3 Real Floquet Theory and Direct Integrals 257
5.4 The Discriminant and Complex Floquet Theory 263
5.5 Potential Theory, Equilibrium Measures, the DOS, and the Lyapunov Exponent 283
5.6 Approximation by Periodic Spectra, I. Finite Gap Sets 306
5.7 Chebyshev Polynomials 312
5.8 Approximation by Periodic Spectra, II. General Sets 319
5.9 Regularity: An Aside 323
5.10 The CD Kernel for Periodic Jacobi Matrices 327
5.11 Asymptotics of the CD Kernel: OPRL on General Sets 334
5.12 Meromorphic Functions on Hyperelliptic Surfaces 344
5.13 Minimal Herglotz Functions and Isospectral Tori 360
Appendix to Section 5.13: A Child's Garden of Almost
Periodic Functions 371
5.14 Periodic OPUC 377
Chapter 6. Toda Flows and Symplectic Structures 379
6.1 Overview 379
6.2 Symplectic Dynamics and Completely Integrable Systems 382
6.3 QR Factorization 387
6.4 Poisson Brackets of OPs, Eigenvalues, and Weights 390
6.5 Spectral Solution and Asymptotics of the Toda Flow 398
6.6 Lax Pairs 403
6.7 The Symes-Deift-Li-Tomei Integration: Calculation of the Lax Unitaries 404
6.8 Complete Integrability of Periodic Toda Flow and Isospectral Tori 408
6.9 Independence of Toda Flows and Trace Gradients 413
6.10 Flows for OPUC 416
Chapter 7. Right Limits 418
7.1 Overview 418
7.2 The Essential Spectrum 419
7.3 The Last-Simon Theorem on A.C. Spectrum 426
7.4 Remling's Theorem on A.C. Spectrum 431
7.5 Purely Reflectionless Jacobi Matrices on Finite Gap Sets 452
7.6 The Denisov-Rakhmanov-Remling Theorem 454
Chapter 8. Szego and Killip-Simon Theorems for Periodic OPRL 456
8.1 Overview 456
8.2 The Magic Formula 457
8.3 The Determinant of the Matrix Weight 460
8.4 A Shohat-Nevai Theorem for Periodic Jacobi Matrices 463
8.5 Controlling the L2 Approach to the Isospectral Torus 465
8.6 A Killip-Simon Theorem for Periodic Jacobi Matrices 473
8.7 Sum Rules for Periodic OPUC 475
Chapter 9. Szego's Theorem for Finite Gap OPRL 477
9.1 Overview 477
9.2 Fractional Linear Transformations 478
9.3 Möbius Transformations 496
9.4 Fuchsian Groups 505
9.5 Covering Maps for Multiconnected Regions 518
9.6 The Fuchsian Group of a Finite Gap Set 525
9.7 Blaschke Products and Green's Functions 540
9.8 Continuity of the Covering Map 556
9.9 Step-by-Step Sum Rules for Finite Gap Jacobi Matrices 562
9.10 The Szego-Shohat-Nevai Theorem for Finite Gap Jacobi Matrices 564
9.11 Theta Functions and Abel's Theorem 570
9.12 Jost Functions and the Jost Isomorphism 576
9.13 Szego Asymptotics 583
Chapter 10. A.C. Spectrum for Bethe-Cayley Trees 591
10.1 Overview 591
10.2 The Free Hamiltonian and Radially Symmetric Potentials 594
10.3 Coefficient Stripping for Trees 597
10.4 A Step-by-Step Sum Rule for Trees 600
10.5 The Global l2 Theorem 601
10.6 The Local l2 Theorem 603
Bibliography 607
Author Index 641
Subject Index 647
Overview
This book presents a comprehensive overview of the sum rule approach to spectral analysis of orthogonal polynomials, which derives from Gábor Szego's classic 1915 theorem and its 1920 extension. Barry Simon emphasizes necessary and sufficient conditions, and provides mathematical background that until now has been available only in journals. Topics include background from the theory of meromorphic functions on hyperelliptic surfaces and the study of covering maps of the Riemann sphere with a finite number of ...