Orthogonal Polynomials on the Unit Circle: Part 1: Classical Theory

✍ Scribed by Barry Simon

• Publisher: American Mathematial Society
• Year: 2009
• Tongue: English
• Leaves: 494
• Series: Colloquium Publications 54
• Category: Library

✦ Synopsis

This two-part volume gives a comprehensive overview of the theory of probability measures on the unit circle, viewed especially in terms of the orthogonal polynomials defined by those measures. A major theme involves the connections between the Verblunsky coefficients (the coefficients of the recurrence equation for the orthogonal polynomials) and the measures, an analog of the spectral theory of one-dimensional Schrödinger operators.

Among the topics discussed along the way are the asymptotics of Toeplitz determinants (Szegő's theorems), limit theorems for the density of the zeros of orthogonal polynomials, matrix representations for multiplication by z (CMV matrices), periodic Verblunsky coefficients from the point of view of meromorphic functions on hyperelliptic surfaces, and connections between the theories of orthogonal polynomials on the unit circle and on the real line.

Readership: Graduate students and research mathematicians interested in analysis.

✦ Table of Contents

Chapter 1 The Basics
Chapter 2 Szegő's theorem
Chapter 3 Tools for Geronimus' theorem
Chapter 4 Matrix representations
Chapter 5 Baxter's theorem
Chapter 6 The strong Szegő theorem
Chapter 7 Verblunsky coefficients with rapid decay
Chapter 8 The density of zeros

Bibliography
Author index
Subject index