A necessary and sufficient condition for nonnegative product linearization of orthogonal polynomials
type: Item
instance of: scholarly article
title: A necessary and sufficient condition for nonnegative product linearization of orthogonal polynomials
zbMATH Open document ID: 1042.42020
author: Ryszard Szwarc
published in: Constructive Approximation
publication date: 2004-03-07T00:00:00Z
review text: Denote by \(\{p_n\}_{n=0}^{N-1}\) a system of orthogonal polynomials with respect to a probability measure \(\mu\) on the real line. If the support of \(\mu\) is infinite \(N=+\infty\), otherwise \(N\) is the number of points in the support of \(\mu\). \noindent In this paper, the author investigates which \(\{p_n\}\) have the property \[ p_n(x)p_m(x)=\sum c(n,m,k)p_k(x) \] with nonnegative coefficients \(c(n,m,k)\) for every choice of \(n,m\) and \(k\). He shows that the nonnegative product linearization is equivalent to the existence of a family of special matrices with triangular support. The method goes through a discrete hyperpolic boundary value problem associated with the three-term recurrence relation. Such new result provides a unified approach to nonnegative linearization problems for orthogonal polynomial systems. Some examples are given for illustration.
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 47B39
zbMATH DE Number: 2051471
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: linearization coefficients
zbMATH Keywords: nonnegative linearization
reviewed by: Youssèf Ben Cheikh
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2051471