In press; publication expected second half of 2010


Szego's Theorem and Its Descendants:
Spectral Theory for L
2 Perturbations
of Orthogonal Polynomials
by Barry Simon (Caltech)

Click here to order from Princeton University Press

Table of Contents 

Chapter 1. Gems of Spectral Theory

  • What Is Spectral Theory

  • OPRL as a Solution of an Inverse Problem

  • Favard's Theorem, the Spectral Theorem, and the Direct Problem for OPRL

  • Gems of Spectral Theory

  • Sum Rules and the Plancherel Theorem

  • Polya's Conjecture and Szego's Theorem

  • OPUC and Szego's Restatement

  • Verblunsky's Form of Szego's Theorem

  • Back to OPRL: Szego Mapping and the Shohat-Nevai Theorem

  • The Killip-Simon Theorem

  • Perturbations of the Periodic Case

  • Other Gems in the Spectral Theory of OPUC

Chapter 2. Szego's Theorem

  • Statement and Strategy

  • The Szego Integral as an Entropy

  • Caratheodory, Herglotz, and Schur Functions

  • Weyl Solutions

  • Coefficient Stripping, Geronimus' and Verblunsky's Theorems, and Continued Fractions

  • The Relative Szego Function and the Step-by-Step Sum Rule

  • The Proof of Szego's Theorem

  • A Higher-Order Szego Theorem

  • The Szego Function and Szego Asymptotics

  • Asymptotics for Weyl Solutions

  • Additional Aspects of Szego's Theorem

  • The Variational Approach to Szego's Theorem

  • Another Approach to Szego Asymptotics

  • Paraorthogonal Polynomials and Their Zeros

  • Asymptotics of the CD Kernel: Weak Limits

  • Asymptotics of the CD Kernel: Continuous Weights

  • Asymptotics of the CD Kernel: Locally Szego Weights

Chapter 3. The Killip-Simon Theorem

  • Statement and Strategy

  • Weyl Solutions and Coefficient Stripping

  • Meromorphic Herglotz Functions

  • Step-by-Step Sum Rules for OPRL

  • The P2 Sum Rule and the Killip-Simon Theorem

  • An Extended Shohat-Nevai Theorem

  • Szego Asymptotics for OPRL

  • The Moment Problem: An Aside

  • The Krein Density Theorem and Indeterminate Moment Problems

  • The Nevai Class and Nevai Delta Convergence Theorem

  • Asymptotics of the CD Kernel: OPRL on [–2,2]

  • Asymptotics of the CD Kernel: Lubinsky's Second Approach

Chapter 4. Sum Rules and Consequences for Matrix OPs

  • Introduction

  • Basics of MOPRL

  • Coefficient Stripping

  • Step-by-Step Sum Rules of MOPRL

  • A Shohat-Nevai Theorem for MOPRL

  • A Killip-Simon Theorem for MOPRL

Chapter 5.  Periodic OPRL

  • Overview

  • m-Functions and Quadratic Irrationalities

  • Real Floquet Theory and Direct Integrals

  • The Discriminant and Complex Floquet Theory

  • Potential Theory, Equilibrium Measures, the DOS, and the Lyapunov Exponent

  • Approximation by Periodic Spectra, I. Finite Gap Sets

  • Chebyshev Polynomials

  • Approximation by Periodic Spectra, II. General Sets

  • Regularity: An Aside

  • The CD Kernel for Periodic Jacobi Matrices

  • Asymptotics of the CD Kernel: OPRL on General Sets

  • Meromorphic Functions on Hyperelliptic Surfaces

  • Minimal Herglotz Functions and Isospectral Tori
           Appendix: A Child's Garden of Almost Periodic Functions

  • Periodic OPUC

Chapter 6.  Toda Flows and Symplectic Structures

  • Overview

  • Symplectic Dynamics and Completely Integrable Systems

  • QR Factorization

  • Poisson Brackets of OPs, Eigenvalues, and Weights

  • Spectral Solution and Asymptotics of the Toda Flow

  • Lax Pairs

  • The Symes--Deift--Li--Tomei Integration: Calculation of the Lax Unitaries

  • Complete Integrability of Periodic Toda Flow and Isospectral Tori

  • Independence of Toda Flows and Trace Gradients

  • Flows for OPUC

Chapter 7.  Right Limits

  • Overview

  • The Essential Spectrum

  • The Last-Simon Theorem on A.C. Spectrum

  • Remling's Theorem on A.C. Spectrum

  • Purely Reflectionless Jacobi Matrices on Finite Gap Sets

  • The Denisov-Rakhmanov--Remling Theorem

Chapter 8.  Szego and Killip-Simon Theorems for Periodic OPRL

  • Overview

  • The Magic Formula

  • The Determinant of the Matrix Weight

  • A Shohat-Nevai Theorem for Periodic Jacobi Matrices

  • Controlling the 2 Approach to the Isospectral Torus

  • A Killip-Simon Theorem for Periodic Jacobi Matrices

  • Sum Rules for Periodic OPUC

Chapter 9.  Szego's Theorem for Finite Gap OPRL

  • Overview

  • Fractional Linear Transformations

  • Mobius Transformations

  • Fuchsian Groups

  • Covering Maps for Multiconnected Regions

  • The Fuchsian Group of a Finite Gap Set

  • Blaschke Products and Green's Functions

  • Continuity of the Covering Map

  • Step-by-Step Sum Rules for Finite Gap Jacobi Matrices

  • The Szego-Shohat-Nevai Theorem for Finite Gap Jacobi Matrices

  • Theta Functions and Abel's Theorem

  • Jost Functions and the Jost Isomorphism

  • Szego Asymptotics

Chapter 10.  A.C. Spectrum for Bethe-Cayley Trees

  • Overview

  • The Free Hamiltonian and Radially Symmetric Potentials

  • Coefficient Stripping for Trees

  • A Step-by-Step Sum Rule for Trees

  • The Global 2 Theorem

  • The Local 2 Theorem

Author Index
Subject Index



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