A note on the interlacing of zeros and orthogonality
type: Item
instance of: scholarly article
title: A note on the interlacing of zeros and orthogonality
zbMATH Open document ID: 1184.42020
published in: Journal of Approximation Theory
publication date: 2010-01-08T00:00:00Z
review text: The author considers a sequence \(\{t_n\}_{n=0}^{\infty}\) of real monic polynomials, with \(\deg(t_n)=n\) for each \(n\), satisfying the property that the zeros of \(t_n\) are real and simple and two consecutive polynomials have no common zeros for any \(n \in \mathbb{N}.\) It is also assumed that the ratio \(\frac{t_{n+1}}{t_{n-1}}\) takes the same value at the zeros of \(t_n\). Under these assumptions, two necessary and sufficient conditions for the orthogonality of \(\{t_n\}\) are obtained. The conditions are given in terms of the interlacing properties of the zeros of \(t_n\) and \(t_{n+1}\) and in terms of the positivity of the ratios above introduced.
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 5655034
zbMATH Keywords: orthogonality of a monic polynomial sequence
zbMATH Keywords: interlacing of zeros
author: Kathy Driver
MaRDI profile type: Publication
full work available at URL: j.jat.2008.11.008
OpenAlex ID: W2052128183
cites work: The zeros of linear combinations of orthogonal polynomials
cites work: Q3032769
cites work: On Orthogonal Polynomials
DOI: 10.1016/J.JAT.2008.11.008
description: scientific article; zbMATH DE number 5655034