Orthogonal polynomials on the unit circle, Part 2, spectral theory / Barry Simon

Auteur principal : Simon, Barry, 1946-, AuteurType de document : MonographieCollection : Colloquium publications, 54Langue : anglais.Pays: Etats Unis.Éditeur : Providence : American Mathematical Society, 2005Description : 1 vol. (xxi-467-1044 p.) ; 26 cmISBN: 9780821836750.ISSN: 0065-9258.Bibliographie : Bibliogr. p. 983-1029. Index.Sujet MSC : 42C05, Nontrigonometric harmonic analysis, Orthogonal functions and polynomials, general theory
47B35, Operator theory - Special classes of linear operators, Toeplitz operators, Hankel operators, Wiener-Hopf operators
30C85, Functions of a complex variable - Geometric function theory, Capacity and harmonic measure in the complex plane
42A10, Harmonic analysis on Euclidean spaces, in one variable, Trigonometric approximation
30H10, Functions of a complex variable - Spaces and algebras of analytic functions, Hardy spaces
En-ligne : Zentralblatt | MathSciNet Item type: Monographie
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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 2 develops some more advanced topics of the OPUC theory. The author adopts the technique from the spectral theory of Schrödinger operators and Jacobi matrices to study the fine structure (absolutely continuous and singular components) of orthogonality measures on the unit circle based on the behavior of their Verblunsky coefficients. Chapter 9 deals with one of the top points of the modern OPUC theory – Rakhmanov’s theorem – and its extensions due to Máté-Nevai-Totik, Khrushchev and Barrios-Lopéz. Various techniques of the spectral analysis are exhibited in Chapter 10. The bulk of Chapter 11 concerns an extremely beautiful theory of periodic Verblunsky coefficients and is very close to results for one-dimensional periodic Schrödinger operators. The key player here is meromorphic functions on hyperelliptic surfaces. Other topics addressed in this volume are the spectral analysis of specific classes of Verblunsky coefficients (sparse, random, subshifts etc.) as well as connections to Jacobi matrices and orthogonal polynomials on the real line. This completes with a reader’s guide (topics and formulae) and a list of conjectures and open questions. 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. (Zentralblatt)

Bibliogr. p. 983-1029. Index

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