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. There is a major theme that 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 Schrodinger operators. Among the topics discussed along the way are the asymptotics of Toeplitz determinants (Szego'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.
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Für höhere Schule und Studium
Für Beruf und Forschung
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ISBN-13
978-0-8218-3757-3 (9780821837573)
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Schweitzer Klassifikation
Part 1: The Basics Szego's theorem Tools for Geronimus' theorem Matrix representations Baxter's theorem The strong Szego theorem Verblunsky coefficients with rapid decay The density of zeros Bibliography Author index Subject index Part 2: 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.
