Akhiezer's orthogonal polynomials and Bernstein-Szegő method for a circular arc
type: Item
instance of: scholarly article
title: Akhiezer's orthogonal polynomials and Bernstein-Szegő method for a circular arc
zbMATH Open document ID: 0916.42016
published in: Journal of Approximation Theory
publication date: 1999-07-14T00:00:00Z
review text: An asymptotic result for polynomials orthogonal on a circular arc was stated without proof in a short note by \textit{N. I. Akhiezer} [Dokl. Akad. Nauk SSSR 130, 247-250 (1960); English transl. in Soviet Math. Dokl. 1, 31-34 (1960; Zbl 0129.28305)]. Although a rather general theory has been developed since then, the results and the methods applied there are still worth studying. The main goal of this paper is to present a rigorous and detailed exposition of Akhiezer's results, and to outline some possible extensions of the method. As a first step, a special class of weights on the arc is considered, for which both the orthogonal polynomials and the outer functions can be constructed explicitly, which yields the asymptotics. Then, a general weight on the arc is approximated by means of the Bernstein-Szegő's method, leading to the desired asymptotic formulas.
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 1230423
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: strong asymptotics
zbMATH Keywords: Bernstein polynomials
author: Leonid Golinskii
reviewed by: Andrei Martínez Finkelshtein
MaRDI profile type: Publication
full work available at URL: jath.1998.3197
OpenAlex ID: W2074276198
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DOI: 10.1006/JATH.1998.3197
description: scientific article; zbMATH DE number 1230423