Asymptotics for orthogonal polynomials
type: Item
instance of: scholarly article
title: Asymptotics for orthogonal polynomials
zbMATH Open document ID: 0617.42014
author: Walter Van Assche
published in: Lecture Notes in Mathematics
publication date: 1987-01-01T00:00:00Z
review text: It is indeed a long time ago - as the author states in the preface - that a textbook on general orthogonal polynomials (on the real line) appeared. Aport from the by now classical books by \textit{G. Szegö} [Orthogonal Polynomials, Am. Math. Soc. Coll. Publ. 23 (1939; Zbl 0023.21505)] and \textit{G. Freud} [Orthogonal Polynomials (1966; Zbl 0169.080)] there are very few new contributions in textbook form. Of course one has the book by \textit{T. S. Chihara} [An introduction to orthogonal polynomials (1978; Zbl 0389.33008)] and the monographs by \textit{Ya. L. Geronimus} [Polynomials orthogonal on a circle and an interval (1960; Zbl 0093.265)] and \textit{P. G. Nevai} [Orthogonal Polynomials, Mem. Am. Math. Soc. 213 (1979; Zbl 0405.33009)]. These three references, however, show the predicament the specialist in orthogonal polynomials finds himself in nowadays: the interest in the subject has caused an enormous expansion - both in scope and in depth - that it is hardly a matter of one author (and much less of one volume) to cover the entire field, including the most recent results. If we now turn to the subject of asymptotics for orthogonal polynomials and make the restriction to those cases where the support of the measure lies on the real axis only, we still have some ground to cover (just think of what has been done with so-called Freud- weights over the last few years). Nevertheless there is a book that covers it all: the lecture note under review. Leaving out just enough of the (sordid) details, the author has succeeded in writing a book that tells the ''orthogonality adept'' everything he alsways wanted to know. After taking a preliminary course in measure theory and potential theory, it is this monograph that finally gives a textbook-like treatment of the asymptotics for orthogonal polynomials. It might be a idea to consider it for use in graduate course in the future. Let us now cover the contents of the book. After a short introduction (including historical r…
reviewed by: Marcel G. de Bruin
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 42-02
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 22-02
Mathematics Subject Classification ID: 40A15
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 39A10
Mathematics Subject Classification ID: 41A60
Mathematics Subject Classification ID: 60J80
zbMATH DE Number: 4000674
zbMATH Keywords: asymptotics for orthogonal polynomials
zbMATH Keywords: compact support
zbMATH Keywords: capacity
zbMATH Keywords: Szegö-class
zbMATH Keywords: Julia-sets
zbMATH Keywords: periodic coefficients
zbMATH Keywords: index
zbMATH Keywords: analytic solutions
zbMATH Keywords: strong asymptotics
zbMATH Keywords: weak asymptotics
zbMATH Keywords: Jacobi polynomials
zbMATH Keywords: Laguerre polynomials
zbMATH Keywords: Hermite polynomials
zbMATH Keywords: Charlier polynomials
zbMATH Keywords: Meixner polynomials
zbMATH Keywords: weight functions
zbMATH Keywords: Plancherel-Rotach asymptotics
zbMATH Keywords: weighted zero distributions
zbMATH Keywords: Christoffel functions
zbMATH Keywords: Toeplitz matrices
zbMATH Keywords: applications
zbMATH Keywords: discrete Schrödinger equations
zbMATH Keywords: birth-death processes
MaRDI profile type: Publication
DOI: 10.1007/BFB0081880
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description: scientific article; zbMATH DE number 4000674