When do linear combinations of orthogonal polynomials yield new sequences of orthogonal polynomials?
type: Item
instance of: scholarly article
title: When do linear combinations of orthogonal polynomials yield new sequences of orthogonal polynomials?
zbMATH Open document ID: 1187.33004
author: M. Alfaro
author: Francisco Marcellán
author: A. Peña
author: M. L. Rezola
published in: Journal of Computational and Applied Mathematics
publication date: 2009-12-18T00:00:00Z
full work available at URL: https://arxiv.org/abs/0711.1740
review text: Let \(\{P_n\}_{n\geq0}\) be a sequence of monic orthogonal polynomials with respect to a linear functional \(u\). The authors tackle linear combinations with constant coefficients \(Q_n=P_n+a_1P_{n-1}+\dots +a_kP_{n-k}\), \(n>k\). The main result provides necessary and sufficient conditions for the sequence \(\{Q_n\}\) to be orthogonal with respect to a linear functional \(v\). The authors give a simple algorithm to compute a polynomial \(h_k\) for which \(u=h_k v\). In the case \(k=2\) a complete description of all families \(\{P_n\}_{n\geq0}\) is obtained such that \(Q_n\) are orthogonal polynomials. Examples of such families are given.
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 5650061
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: recurrence relations
zbMATH Keywords: linear functionals
zbMATH Keywords: Chebyshev polynomials
zbMATH Keywords: Jacobi matrices
reviewed by: Leonid Golinskii
MaRDI profile type: Publication
OpenAlex ID: W2129486468
arXiv ID: 0711.1740
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DOI: 10.1016/J.CAM.2009.02.060
description: scientific article