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

Simon, Barry

ISBN 10: 0821848631 / ISBN 13: 9780821848630

Editore: American Mathematial Society, 2009

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Dati bibliografici

• Titolo: Orthogonal Polynomials on the Unit Circle : Part 1: Classical Theory
• Editore: American Mathematial Society
• Data di pubblicazione: 2009
• Lingua: Inglese
• ISBN 10: 0821848631
• ISBN 13: 9780821848630
• Rilegatura: Brossura
• Condizione: New

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Riassunto

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 (Szeg '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.