Asymptotic properties of balanced extremal Sobolev polynomials: Coherent case
type: Item
instance of: scholarly article
title: Asymptotic properties of balanced extremal Sobolev polynomials: Coherent case
zbMATH Open document ID: 0942.42015
published in: Journal of Approximation Theory
publication date: 1999-12-20T00:00:00Z
review text: Consider a Sobolev inner product of the form \[ (p,q)=\int pq d\mu_0+\int p'q' d\mu_1. \] If \(\{P_n\}\) and \(\{Q_n\}\) denote the sequences of monic orthogonal polynomials with respect to the measures \(d\mu_0\) and \(d\mu_1\), respectively, then \((d\mu_0,d\mu_1)\) is called a coherent pair of measures if there exist non-zero constants \(D_n\) such that \[ Q_n=\frac{P_{n+1}'}{n+1}+D_n\frac{P_{n-1}'}{n-1},\quad n\geq 2. \] In [J. Approximation Theory 89, No. 3, 321-343 (1997; Zbl 0880.42012)] \textit{H. G.~Meijer} classified all coherent pairs of measures. In the present paper this result is used to study the case that \(\text{supp}(d\mu_0)=[-1,1]\). In that case the asymptotic properties of the Sobolev orthogonal polynomials are studied. Also the behaviour of the norms and the zeros of these polynomials is discussed. It is shown that in some case these Sobolev orthogonal polynomials asymptotically behave like the monic polynomial sequence of polynomials which is orthogonal with respect to a new measure being a linear combination of the measures \(d\mu_0\) and \(d\mu_1\). The authors conjecture that this result is still valid in a more general setting.
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 1356330
zbMATH Keywords: Sobolev orthogonal polynomials
zbMATH Keywords: asymptotics
zbMATH Keywords: coherent pairs of measures
author: M. Alfaro
author: Andrei Martínez Finkelshtein
author: M. L. Rezola
reviewed by: Roelof Koekoek
MaRDI profile type: Publication
full work available at URL: jath.1998.3336
OpenAlex ID: W2013486162
DOI: 10.1006/JATH.1998.3336
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description: scientific article; zbMATH DE number 1356330