The strong Chebyshev distribution and orthogonal Laurent polynomials
type: Item
instance of: scholarly article
title: The strong Chebyshev distribution and orthogonal Laurent polynomials
zbMATH Open document ID: 0930.42011
author: S. Clement Cooper
author: Philip E. Gustafson
published in: Journal of Approximation Theory
publication date: 2000-02-06T00:00:00Z
review text: This is a worked out example of a sequence of orthogonal Laurent polynomials, based on the Chebyshev polynomials. The strong Chebyshev distribution is a (symmetric) weight on \([-b,-a] \cup [a,b]\) for which all the moments exist, even for negative indices. It is obtained from the Chebyshev weight \(1/\sqrt{1-x^2}\) on \([-1,1]\) by a quadratic transformation. The corresponding Chebyshev Laurent polynomials are obtained from the Chebyshev polynomials of the first kind by using a rational change of variable. This allows the authors to explicitly find the orthogonality, the recurrence relation, and closed form expressions. The (uniform) convergence of the continued fraction associated with these Laurent polynomials is investigated. Finally, the zeros of the Chebyshev-Laurent polynomials are given explicitly. \textit{B. A. Hagler, W. B. Jones} and \textit{W. J. Thron} [Lect. Notes Pure Appl. Math. 199, 187-208 (1998)] have used the same idea to find orthogonal Laurent polynomials of Jacobi, Hermite and Laguerre types.
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 1197037
zbMATH Keywords: Laurent polynomials
zbMATH Keywords: Chebyshev polynomials
zbMATH Keywords: strong moment problem
reviewed by: Walter Van Assche
MaRDI profile type: Publication
full work available at URL: jath.1997.3161
OpenAlex ID: W2030877523
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DOI: 10.1006/JATH.1997.3161
description: scientific article; zbMATH DE number 1197037