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Orthogonal Polynomials on the Unit Circle: Part 2: Spectral Theory
Barry Simon, California Institute of Technology, Pasadena, CA

Colloquium Publications
2005; 578 pp; softcover
Volume: 54
ISBN-10: 0-8218-4864-X
ISBN-13: 978-0-8218-4864-7
List Price: US$104
Member Price: US$83.20
Order Code: COLL/54.2.S
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Winner of the 2015 Bolyai Prize of the Hungarian Academy of Sciences!

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.

The book is suitable for graduate students and researchers interested in analysis.


Graduate students and research mathematicians interested in analysis.


"This completes with a reader's guide (topics and formulae) and a list of conjectures and open questions. Detailed historic and bibliographic notes are appended to each chapter. A reader is furnished with an extensive notation list and an exhaustive bibliography. The book will be of interest to a wide range of mathematicians."

-- Zentralblatt MATH

"The author has now opened new vistas with a monumental treatise that integrates old and new aspects of the subject and describes important connections with mathematical physics, especially with spectral theory of Schrödinger Operations. ... The author has made every effort to guide the reader toward a clear understanding of the material and its various interconnections. The result is an instant classic that will become a standard source for novices and veterans alike."

-- Mathematical Reviews

Table of Contents

  • Rakhmanov's theorem and related issues
  • Techniques of spectral analysis
  • Periodic Verblunsky coefficients
  • Spectral analysis of specific classes of Verblunsky coefficients
  • The connection to Jacobi matrices
  • Reader's guide: Topics and formulae
  • Perspectives
  • Twelve great papers
  • Conjectures and open questions
  • Bibliography
  • Author index
  • Subject index
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