Colloquium Publications
Volume: 54;
2004;
1044 pp;
Softcover
MSC: Primary 42; 05; 34;
Secondary 47; 30
Print ISBN: 978-0-8218-4867-8
Product Code: COLL/54.S
List Price: $169.00
AMS Member Price: $135.20
MAA Member Price: $152.10
Item(s) contained in this set are available for individual sale:
Supplemental Materials
Orthogonal Polynomials on the Unit Circle: Part 1: Classical Theory; Part 2: Spectral Theory
Share this pageBarry Simon
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.
Reviews & Endorsements
Simon's work is not just a book about orthogonal polynomials but also about probability measures on one-dimensional Schrödinger operators and operator theory. It is extremely complex, multilayered, fascinating, and inspiring, while remaining very readable (even for advanced students). Without a doubt this monograph will become the standard reference for the theory of orthogonal polynomials on the unit circle for a long time to come.
-- Jahresbericht der DMV
Undoubtedly that ... this book will become a standard reference in the field tracing the way for future investigations on orthogonal polynomials and their applications. Combining methods from various areas of analysis (calculus, real analysis, functional analysis, complex analysis) as well as by the importance of the orthogonal pholynomials in applications, the book will have a large audience including researchers in mathematics, physics, (and) engineering.
-- Stefan Cobzas, Studia Universitatis Babes-Bolyai, Mathematica