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Szegő's Theorem and Its Descendants

Book Description

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.

Table of Contents

  1. Cover
  2. Halftitle
  3. Title
  4. Copyright
  5. Contents
  6. Preface
  7. Chapter 1. Gems of Spectral Theory
    1. 1.1 What Is Spectral Theory?
    2. 1.2 OPRL as a Solution of an Inverse Problem
    3. 1.3 Favard’s Theorem, the Spectral Theorem, and the Direct Problem for OPRL
    4. 1.4 Gems of Spectral Theory
    5. 1.5 Sum Rules and the Plancherel Theorem
    6. 1.6 Pólya’s Conjecture and Szegő’s Theorem
    7. 1.7 OPUC and Szegő’s Restatement
    8. 1.8 Verblunsky’s Form of Szegő’s Theorem
    9. 1.9 Back to OPRL: Szegő Mapping and the Shohat-Nevai Theorem
    10. 1.10 The Killip-Simon Theorem
    11. 1.11 Perturbations of the Periodic Case
    12. 1.12 Other Gems in the Spectral Theory of OPUC
  8. Chapter 2. Szegő’s Theorem
    1. 2.1 Statement and Strategy
    2. 2.2 The Szegő Integral as an Entropy
    3. 2.3 Carathéodory, Herglotz, and Schur Functions
    4. 2.4 Weyl Solutions
    5. 2.5 Coefficient Stripping, Geronimus’ and Verblunsky’s Theorems, and Continued Fractions
    6. 2.6 The Relative Szegő Function and the Step-by-Step Sum Rule
    7. 2.7 The Proof of Szegő’s Theorem
    8. 2.8 A Higher-Order Szegő Theorem
    9. 2.9 The Szegő Function and Szegő Asymptotics
    10. 2.10 Asymptotics for Weyl Solutions
    11. 2.11 Additional Aspects of Szegő’s Theorem
    12. 2.12 The Variational Approach to Szegő’s Theorem
    13. 2.13 Another Approach to Szegő Asymptotics
    14. 2.14 Paraorthogonal Polynomials and Their Zeros
    15. 2.15 Asymptotics of the CD Kernel: Weak Limits
    16. 2.16 Asymptotics of the CD Kernel: Continuous Weights
    17. 2.17 Asymptotics of the CD Kernel: Locally Szegő Weights
  9. Chapter 3. The Killip-Simon Theorem: Szegő for OPRL
    1. 3.1 Statement and Strategy
    2. 3.2 Weyl Solutions and Coefficient Stripping
    3. 3.3 Meromorphic Herglotz Functions
    4. 3.4 Step-by-Step Sum Rules for OPRL
    5. 3.5 The P2 Sum Rule and the Killip-Simon Theorem
    6. 3.6 An Extended Shohat-Nevai Theorem
    7. 3.7 Szegő Asymptotics for OPRL
    8. 3.8 The Moment Problem: An Aside
    9. 3.9 The Krein Density Theorem and Indeterminate Moment Problems
    10. 3.10 The Nevai Class and Nevai Delta Convergence Theorem
    11. 3.11 Asymptotics of the CD Kernel: OPRL on [-2, 2]
    12. 3.12 Asymptotics of the CD Kernel: Lubinsky’s Second Approach
  10. Chapter 4. Sum Rules and Consequences for Matrix Orthogonal Polynomials
    1. 4.1 Introduction
    2. 4.2 Basics of MOPRL
    3. 4.3 Coefficient Stripping
    4. 4.4 Step-by-Step Sum Rules of MOPRL
    5. 4.5 A Shohat-Nevai Theorem for MOPRL
    6. 4.6 A Killip-Simon Theorem for MOPRL
  11. Chapter 5. Periodic OPRL
    1. 5.1 Overview
    2. 5.2 m-Functions and Quadratic Irrationalities
    3. 5.3 Real Floquet Theory and Direct Integrals
    4. 5.4 The Discriminant and Complex Floquet Theory
    5. 5.5 Potential Theory, Equilibrium Measures, the DOS, and the Lyapunov Exponent
    6. 5.6 Approximation by Periodic Spectra, I. Finite Gap Sets
    7. 5.7 Chebyshev Polynomials
    8. 5.8 Approximation by Periodic Spectra, II. General Sets
    9. 5.9 Regularity: An Aside
    10. 5.10 The CD Kernel for Periodic Jacobi Matrices
    11. 5.11 Asymptotics of the CD Kernel: OPRL on General Sets
    12. 5.12 Meromorphic Functions on Hyperelliptic Surfaces
    13. 5.13 Minimal Herglotz Functions and Isospectral Tori
    14. Appendix to Section 5.13: A Child’s Garden of Almost Periodic Functions
    15. 5.14 Periodic OPUC
  12. Chapter 6. Toda Flows and Symplectic Structures
    1. 6.1 Overview
    2. 6.2 Symplectic Dynamics and Completely Integrable Systems
    3. 6.3 QR Factorization
    4. 6.4 Poisson Brackets of OPs, Eigenvalues, and Weights
    5. 6.5 Spectral Solution and Asymptotics of the Toda Flow
    6. 6.6 Lax Pairs
    7. 6.7 The Symes-Deift-Li-Tomei Integration: Calculation of the Lax Unitaries
    8. 6.8 Complete Integrability of Periodic Toda Flow and Isospectral Tori
    9. 6.9 Independence of Toda Flows and Trace Gradients
    10. 6.10 Flows for OPUC
  13. Chapter 7. Right Limits
    1. 7.1 Overview
    2. 7.2 The Essential Spectrum
    3. 7.3 The Last-Simon Theorem on A.C. Spectrum
    4. 7.4 Remling’s Theorem on A.C. Spectrum
    5. 7.5 Purely Reflectionless Jacobi Matrices on Finite Gap Sets
    6. 7.6 The Denisov-Rakhmanov-Remling Theorem
  14. Chapter 8. Szegő and Killip-Simon Theorems for Periodic OPRL
    1. 8.1 Overview
    2. 8.2 The Magic Formula
    3. 8.3 The Determinant of the Matrix Weight
    4. 8.4 A Shohat-Nevai Theorem for Periodic Jacobi Matrices
    5. 8.5 Controlling the ℓ2 Approach to the Isospectral Torus
    6. 8.6 A Killip-Simon Theorem for Periodic Jacobi Matrices
    7. 8.7 Sum Rules for Periodic OPUC
  15. Chapter 9. Szegő’s Theorem for Finite Gap OPRL
    1. 9.1 Overview
    2. 9.2 Fractional Linear Transformations
    3. 9.3 Möbius Transformations
    4. 9.4 Fuchsian Groups
    5. 9.5 Covering Maps for Multiconnected Regions
    6. 9.6 The Fuchsian Group of a Finite Gap Set
    7. 9.7 Blaschke Products and Green’s Functions
    8. 9.8 Continuity of the Covering Map
    9. 9.9 Step-by-Step Sum Rules for Finite Gap Jacobi Matrices
    10. 9.10 The Szegő-Shohat-Nevai Theorem for Finite Gap Jacobi Matrices
    11. 9.11 Theta Functions and Abel’s Theorem
    12. 9.12 Jost Functions and the Jost Isomorphism
    13. 9.13 Szegő Asymptotics
  16. Chapter 10. A.C. Spectrum for Bethe-Cayley Trees
    1. 10.1 Overview
    2. 10.2 The Free Hamiltonian and Radially Symmetric Potentials
    3. 10.3 Coefficient Stripping for Trees
    4. 10.4 A Step-by-Step Sum Rule for Trees
    5. 10.5 The Global ℓ2 Theorem
    6. 10.6 The Local ℓ2 Theorem
  17. Bibliography
  18. Author Index
  19. Subject Index