On some classes of polynomials orthogonal on arcs of the unit circle connected with symmetric orthogonal polynomials on an interval
type: Item
instance of: scholarly article
title: On some classes of polynomials orthogonal on arcs of the unit circle connected with symmetric orthogonal polynomials on an interval
zbMATH Open document ID: 0921.42017
author: Alexei Zhedanov
published in: Journal of Approximation Theory
publication date: 1999-09-21T00:00:00Z
review text: In this paper, the author uses the relationship between orthogonal polynomials on the unit circle and the real line to derive interesting new results for the unit circle case. Consider the recurrence relation \[ S_{n+1}(x)= xS_n(x)-S_{n-1}(x), \] with \(S_0(x) :=1\), \(S_1(x)=x\) and \(u_n>0\) \(\forall n\). This generates a sequence \(\{S_n\}\) of monic polynomials that are orthogonal with respect to a symmetric measure \(\mu\) on a symmetric real interval \([-2c,2c]\): \[ \int^{2c}_{-2c}S_nS_md\mu=h_n\delta_{nm} (h_n>0). \] Fix a scaling parameter \(d>0\), and relate \(z\) and \(x\) by \(x=d(z^{1/2}+z^{-1/2})\). For \(n\geq 0\), let \[ P_n(z):={d^{-n-1}z^{n/2}\Bigl(z^{1/2}S_{n+1}(x)-{S_{n+1}(2d) \over S_n(2d)}S_n(x)\Bigr)\over z-1}. \] This is the DG (Delsarte-Genin) mapping of \(\{S_n\}\) with parameter \(d\). It is known that the \(\{P_n\}\) satisfy a recurrence of the form \[ P_{n+1}(z)=zP_n(z)-a_nP^*_n(z) \] where \(P_0(z)= P^*_0 (z)=1\), \(| a_n|<1\) and \(P^*_n(z): =z^n\overline {P_n(1/ \overline z})\). Consequently the \(\{P_n\}\) are orthogonal with respect to a measure on the unit circle: \[ \int^{2\pi}_0 P_n\overline {P_m}d\sigma =\kappa_n \delta_{mn} (\kappa_n >0). \] The author shows how different choices of \(d\) may be used to ``squeeze'' orthogonal polynomials on the circle, yielding new systems of orthogonal polynomials. Moreover, he shows how the DG mapping yields sieved polynomials, both on the circle and the interval, and also circle analogues of the Askey-Wilson polynomials. Finally, some \(q\)-orthogonal polynomials are discussed, as well as the relation of DG to chain sequences and to Bauer's \(g\)-algorithm.
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 30C10
Mathematics Subject Classification ID: 30B70
zbMATH DE Number: 1214033
zbMATH Keywords: Delsarte-Genin mapping
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: sieved polynomials
zbMATH Keywords: Askey-Wilson polynomials
reviewed by: Doron S. Lubinsky
MaRDI profile type: Publication
full work available at URL: jath.1998.3179
OpenAlex ID: W1975520015
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DOI: 10.1006/JATH.1998.3179
description: scientific article; zbMATH DE number 1214033