Asymptotic behaviour of orthogonal polynomials on the unit circle with asymptotically periodic reflection coefficients
type: Item
instance of: scholarly article
title: Asymptotic behaviour of orthogonal polynomials on the unit circle with asymptotically periodic reflection coefficients
zbMATH Open document ID: 0889.42021
author: Franz Peherstorfer
author: R. Steinbauer
published in: Journal of Approximation Theory
publication date: 1998-06-01T00:00:00Z
review text: Monic orthogonal polynomials on the unit circle are completely characterized by the Szegő recurrence relation \(P_{n+1}(z) = zP_n(z) - \bar{b_n} P_n^*(z)\), where \(b_n\) \((n=0,1,2,\ldots)\) are known as the reflection coefficients (Schur parameters) and \(P_n^*(z) = z^n \bar{P}_n(1/z)\) are the reversed polynomials. In the present paper the authors investigate the situation when \(\lim_{n \to \infty} b_{nN+j} = a_j\), where \(a_{j+N}=a_j\). First they investigate the polynomials with the periodic reflection coefficients \(a_n\) \((n=0,1,2,\ldots)\) and then they study the asymptotic periodic case as a perturbation of the periodic case. They obtain comparative asymptotics on and off the support of the orthogonality measure (which is essentially the union of several arcs on the unit circle) and give a description of the (asymptotic) location of the zeros. Finally, they also have results regarding the absolutely continuous part of the measure and they show that there are at most finitely many mass points if \(b_n-a_n\) converges geometrically fast to zero.
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 30C10
zbMATH DE Number: 1006709
zbMATH Keywords: orthogonal polynomials on the unit circle
zbMATH Keywords: reflection coefficients
zbMATH Keywords: asymptotic periodicity
reviewed by: Walter Van Assche
MaRDI profile type: Publication
full work available at URL: jath.1996.3026
OpenAlex ID: W2013796135
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DOI: 10.1006/JATH.1996.3026
description: scientific article; zbMATH DE number 1006709