Coherent pairs and zeros of Sobolev-type orthogonal polynomials
type: Item
instance of: scholarly article
title: Coherent pairs and zeros of Sobolev-type orthogonal polynomials
zbMATH Open document ID: 0784.33004
DOI: 10.1016/0019-3577(93)90037-Y
published in: Indagationes Mathematicae. New Series
publication date: 1993-12-06T00:00:00Z
review text: In the paper being reviewed the author deals, for fixed \(\lambda\geq 0\), with the set \(\{P^ \lambda_ n\}^ \infty_{n=0}\) of polynomials which are orthogonal with respect to a nondiscrete Sobolev inner product of the form \[ \langle f,g\rangle:=\int^ b_ af(x)g(x)d\psi_ 0(x)+\lambda\int^ b_ 0f'(x)g'(x)d\psi_ 1(x), \] where \(-\infty\leq a<b\leq\infty\) and \(\psi_ 0\), \(\psi_ 1:(a,b)\to\mathbb{R}\) are suitable distribution functions. Up to new results on the zeros of \(P^ \lambda_ n\) were only known for special choices of \(\psi_ 0\) and \(\psi_ 1\). For example, in the case \(d\psi_ 0(x)=d\psi_ 1(x)=dx\) it was proved that, for sufficiently large \(\lambda\), the zeros of \(P^ \lambda_ n\) are real and interlace with the zeros of the \((n-1)\)-th Legendre polynomial. But using the concept of coherent and symmetrically coherent pairs of measures, the author is able to derive more general results for the zero distribution of \(P^ \lambda_ n\). Specifically, he shows that in the coherent case, for \(n\geq 2\) and sufficiently large \(\lambda\), the zeros of \(P^ \lambda_ n\) are real, simple and interlace with the zeros of \(Q_{n-1}\) and \(R_{n-1}\), where \(\{Q_ n\}^ \infty_{n=0}\) and \(\{R_ n\}^ \infty_{n=0}\) denote sets of polynomials orthogonal with respect to \(d\psi_ 0\), respectively \(d\psi_ 1\).
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 469229
zbMATH Keywords: zero distribution
zbMATH Keywords: coherent pairs of measures
author: Henk Meijer
reviewed by: J. Saurer
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 469229