Connection between orthogonal polynomials on the unit circle and bounded interval
type: Item
instance of: scholarly article
title: Connection between orthogonal polynomials on the unit circle and bounded interval
zbMATH Open document ID: 1066.33009
author: E. Berriochoa
author: A. Cachafeiro
author: José M. García-Amor
published in: Journal of Computational and Applied Mathematics
publication date: 2005-02-22T00:00:00Z
review text: The Chebyshev polynomials \(T_n(x)=\cos(n\arccos x)\) can be written \(T_n(x)=\frac {z^n+z^{-n}}2\) for \(x =\frac {z+z^{-1}}{2}\,.\) The authors consider this type of transformation between measures on a bounded interval and the unit circle due to Szegő in detail. They relate the sequences of orthogonal polynomials with respect to both measures and they also relate the coefficients of the three-term recurrence relation with the Schur parameters. When the measures belong to the Szegő class, they study the asymptotic behavior of the orthogonal polynomials on the interval, outside the support of the measure, as well as inside. They also transform the generalized polynomials and they study the orthogonality properties of the new polynomials, obtaining new interesting results. Finally they solve two inverse problems connected with the transformation studied.
reviewed by: W. Koepf
Mathematics Subject Classification ID: 33C47
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 2136625
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: Jacobi measures
zbMATH Keywords: Szegő's theory
MaRDI profile type: Publication
full work available at URL: j.cam.2004.09.018
OpenAlex ID: W2085514328
cites work: Q4043533
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cites work: Q4170993
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DOI: 10.1016/J.CAM.2004.09.018
description: scientific article; zbMATH DE number 2136625