Orthogonal polynomials on arcs of the unit circle. I
type: Item
instance of: scholarly article
title: Orthogonal polynomials on arcs of the unit circle. I
zbMATH Open document ID: 0861.42014
author: Franz Peherstorfer
author: R. Steinbauer
published in: Journal of Approximation Theory
publication date: 1997-04-23T00:00:00Z
review text: In this paper polynomials \(P_n\) orthogonal on the unit circle with respect to a weight function of the form \(f(\theta)= {\mathcal W}(\theta)/{\mathcal A}(\theta) r(\theta)\) for \(\theta\in E_\ell= \bigcup^\ell_{j=1} [\theta_{2j-1}, \theta_{2j}]\) are investigated. Here \({\mathcal A}\) is a real trigonometric polynomial which has no zeros on \(E_\ell\), \({\mathcal W}\) is a trigonometric polynomial vanishing at some of the endpoints \(\theta_k\) and \(r(\theta)= (-1)^j \sqrt{|{\mathcal R}(\theta)|}\) when \(\theta\in (\theta_{2j-1}, \theta_{2j})\), where \({\mathcal R}\) is the trigonometric polynomial vanishing at all the endpoints \(\theta_k\). Observe that the weight function \(f\) is not necessarily positive on \(E_\ell\). First, a few examples are given (for two intervals), which are useful in understanding the kind of problems considered in this paper. It is shown that \(P_n\) is orthogonal with respect to \(f\) if and only if there exists a polynomial \(Q_m\) and a polynomial \(g_n\) such that \(W(z) P^2_n(z)- V(z) Q^2_m(z)= z^n A(z)g_n(z)\). Furthermore, the polynomial \(Q_m\) can then be represented in terms of the orthogonal polynomials \(P_n\) and the polynomials of the second kind. These orthogonal polynomials are interesting because they are useful explicit examples of orthogonal polynomials on several arcs, in particular polynomials with periodic reflection coefficients (from a certain index onward). The paper is rather technical but deserves close attention.
reviewed by: Walter Van Assche
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 30C10
zbMATH DE Number: 885477
zbMATH Keywords: asymptotically periodic recurrences
zbMATH Keywords: unit circle
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: periodic reflection coefficients
MaRDI profile type: Publication
full work available at URL: jath.1996.0035
OpenAlex ID: W2009526469
DOI: 10.1006/JATH.1996.0035
description: scientific article; zbMATH DE number 885477