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

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. Szegö 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 Szegö 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 Szegö 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 Szegö 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].
[Leonid Golinskii (Kharkov)]

MSC 2000:: *42C05 General theory of orthogonal functions and polynomials
47B35 Toeplitz operators, etc.
30C85 Capacity and harmonic measure in the complex plane
30D55 H (sup p)-classes
42A10 Trigonometric approximation
05E35 Orthogonal polynomials (combinatorics)
34L99 Ordinary differential operators
42-02 Research monographs (Fourier analysis)
33-02 Research monographs (special functions)

Keywords: measures on the unit circle; Verblunsky coefficients; Szegö recurrences; Carathéodory and Schur functions; Szegö theorem; Geronimus theorem; matrix representations; Baxter theorem; density of zeros

Citations: Zbl 1082.42021

Cited in: Zbl 1188.33015 Zbl 1185.34131 Zbl 1164.42017 Zbl 1130.82017 Zbl 1108.37002 Zbl 1134.42017 Zbl 1127.47037 Zbl 1118.47023 Zbl 1117.42005 Zbl 1082.42016 Zbl 1082.42021 Zbl 1086.42501

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