The theorems of Stieltjes and Favard
type: Item
instance of: scholarly article
title: The theorems of Stieltjes and Favard
zbMATH Open document ID: 1232.42023
DOI: 10.1007/BF03321801
published in: Computational Methods and Function Theory
publication date: 2011-10-28T00:00:00Z
review text: A possible way to develop the theory of orthogonal polynomials is to begin with a sequence of polynomials defined by a three-term recurrence relation of the form \(p_0(x)=1\), \(p_1(x)=x-\alpha _0\), \(p_{n+2}(x)=(x-\alpha _{n+1})p_{n+1}(x)-\lambda _np_n(x)\) for \(n\geq 0\), where each \(\alpha _{n+1}\) and \(\lambda _n\) is real, and \(\lambda _n>0\) for \(n=0,1,2,\dots.\) Under these conditions, Favard's theorem asserts that there is a positive measure \(\mu \) such that the obtained polynomials are orthogonal in the sense \(\int _\mathbb{R}p_n(x)\;p_k(x)\;d\mu =0\) for \(n\neq k\). It is well known that the zeros of \(p_{n-1}\) and \(p_n\) are real, simple and interlacing. Beyond this, Stieltjes showed that if \(m<n-1\) then \(p_m\) and \(p_n\) have interlacing zeros. Here the author gives a modern proof of Favard's theorem and another proof of a result of de Boor and Saff which extends Stieltjes' theorem.
reviewed by: Nicolae Cotfas
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 33D45
zbMATH DE Number: 5965039
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: interlacing zeros
zbMATH Keywords: Favard's theorem
author: Alan F. Beardon
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf03321801
OpenAlex ID: W2044928488
description: scientific article; zbMATH DE number 5965039