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Orthogonal polynomials on the unit circle. Part 1: Classical theory. (English) Zbl 1082.42020
Colloquium Publications. American Mathematical Society 54, Part 1. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3446-0/hbk). xxv, 466 p. $ 89.00 (2005).

The two-part treatise by Barry Simon, the world renowned expert in mathematical physics, come out in the same AMS Colloquium Publications series as the celebrated book by G. Szegő on orthogonal polynomials 75 years earlier. The main subject is the theory of probability measures on the unit circle, viewed especially in terms of the orthogonal polynomials defined by those measures. Part 1 begins with a concise preface where the author explains his evolution from the theory of Schrödinger operators and Jacobi matrices to the theory of orthogonal polynomials on the unit circle (OPUC). The first chapter develops the basic notions of the theory: Verblunsky coefficients and the Szegő recurrences, Carathéodory and Schur functions. It also contains a nice collection of examples of OPUC as well as a succinct introduction to operator and spectral theory. In Chapter 2 one of the highlights of the theory - the Szegő theorem - is discussed. Chapter 4 presents two basic matrix representations of the multiplication operator and provides some spectral consequences for the OPUC theory. Chapters 5 and 6 deal with another two fundamental results of the theory: Baxter’s theorem and the strong Szegő theorem. Other topics addressed in the first volume concern measures with exponentially decaying Verblunsky coefficients and the density of zeros. 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.

[See also the review of Part 2: Spectral theory in Zbl 1082.42021].

42C05General theory of orthogonal functions and polynomials
47B35Toeplitz operators, Hankel operators, Wiener-Hopf operators
30C85Capacity and harmonic measure in the complex plane
30D55H (sup p)-classes (MSC2000)
42A10Trigonometric approximation
05E35Orthogonal polynomials (combinatorics) (MSC2000)
34L99Ordinary differential operators
42-02Research monographs (Fourier analysis)
33-02Research monographs (special functions)