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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 ShohatNevai Theorem

The KillipSimon
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 StepbyStep Sum Rule

The Proof of
Szego's Theorem

A HigherOrder
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
KillipSimon Theorem

Statement and
Strategy

Weyl Solutions and
Coefficient Stripping

Meromorphic
Herglotz Functions

StepbyStep Sum
Rules for OPRL

The P_{2}
Sum Rule and the KillipSimon Theorem

An Extended
ShohatNevai 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

StepbyStep Sum
Rules of MOPRL

A ShohatNevai
Theorem for MOPRL

A KillipSimon
Theorem for MOPRL
Chapter 5.
Periodic OPRL

Overview

mFunctions
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 SymesDeiftLiTomei
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 LastSimon
Theorem on A.C. Spectrum

Remling's Theorem
on A.C. Spectrum

Purely
Reflectionless Jacobi Matrices on Finite Gap Sets

The
DenisovRakhmanovRemling Theorem
Chapter 8. Szego
and KillipSimon Theorems for Periodic OPRL

Overview

The Magic Formula

The Determinant of
the Matrix Weight

A ShohatNevai
Theorem for Periodic Jacobi Matrices

Controlling the
ℓ^{2}
Approach to the Isospectral Torus

A KillipℓSimon
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

StepbyStep Sum
Rules for Finite Gap Jacobi Matrices

The
SzegoShohatNevai 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 BetheCayley Trees

Overview

The Free
Hamiltonian and Radially Symmetric Potentials

Coefficient
Stripping for Trees

A StepbyStep Sum
Rule for Trees

The Global
ℓ^{2 }
Theorem

The Local
ℓ^{2 }
Theorem
Bibliography
Author Index
Subject Index 