Szegö polynomials: Quadrature rules on the unit circle and on \([-1,1]\)
type: Item
instance of: scholarly article
title: Szegö polynomials: Quadrature rules on the unit circle and on \([-1,1]\)
zbMATH Open document ID: 1059.41014
author: R. Bressan
author: S. F. Menegasso
author: A. Sri Ranga
published in: Rocky Mountain Journal of Mathematics
publication date: 2004-03-17T00:00:00Z
review text: Instead of the classical Joukwoski transformation mapping the unit circle onto the interval \([-1,1]\), in this paper, the authors make use of the transformation introduced by \textit{P. Delsarte} and \textit{Y. V. Genin} [IEEE Trans. Acoust. Speech Signal Process. 34, 470--478 (1986)], \[ x= x(z)= (z^{1/2}+ z^{-1/2})/2 \] in order to establish some connections between the families of orthogonal polynomials on \([-1,1]\) with respect to two given positive measures \(dp^{(1)}(x)\) and \(dp^{(2)}(x)\) such that \(dp^{(1)}(x)= (1-x^2)dp^{(2)}(x)\) and the Szegő polynomials on the unit circle with respect to the Borel measure \(dm(z)=- dp^{(1)}(x(z))\). Connection is made through the so-called para-orthogonal polynomials. Relations between certain interpolatory quadrature rules with positive weights or coefficients to estimate integrals both on \([-1,1]\) and the unit circle are also investigated.
reviewed by: Pablo González-Vera
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 2060093
zbMATH Keywords: Szegő polynomials
zbMATH Keywords: para-orthogonal polynomials
zbMATH Keywords: Gaussian quadature rules
zbMATH Keywords: Szegő quadrature rules
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181069967
OpenAlex ID: W1984215640
cites work: A connection between quadrature formulas on the unit circle and the interval \([-1,1]\)
cites work: Szegö polynomials: Some relations to \(L\)-orthogonal and orthogonal polynomials
cites work: Szegő polynomials and quadrature formulas on the unit circle
cites work: Q3830387
cites work: Some results about numerical quadrature on the unit circle
cites work: Moment Theory, Orthogonal Polynomials, Quadrature, and Continued Fractions Associated with the unit Circle
cites work: A Strong Stieltjes Moment Problem
cites work: Linear Combinations of Orthogonal Polynomials Generating Positive Quadrature Formulas
cites work: Über die Konvergenz von Quadraturverfahren
cites work: Q4341408
cites work: On some classes of polynomials orthogonal on arcs of the unit circle connected with symmetric orthogonal polynomials on an interval
DOI: 10.1216/RMJM/1181069967
description: scientific article; zbMATH DE number 2060093