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.

Readership

Graduate students and research mathematicians interested in analysis.

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