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Orthogonal Polynomials on the Unit Circle: Part 1: Classical Theory
Barry Simon, California Institute of Technology, Pasadena, CA
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Colloquium Publications
2005; 466 pp; softcover
Volume: 54
ISBN-10: 0-8218-4863-1
ISBN-13: 978-0-8218-4863-0
List Price: US$98
Member Price: US$78.40
Order Code: COLL/54.1.S
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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

  • The Basics
  • Szegő's theorem
  • Tools for Geronimus' theorem
  • Matrix representations
  • Baxter's theorem
  • The strong Szegő theorem
  • Verblunsky coefficients with rapid decay
  • The density of zeros
  • Bibliography
  • Author index
  • Subject index
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