Contracted zero distributions of extremal polynomials associated with slowly decaying weights
type: Item
instance of: scholarly article
title: Contracted zero distributions of extremal polynomials associated with slowly decaying weights
zbMATH Open document ID: 0959.31003
published in: Constructive Approximation
publication date: 2000-11-05T00:00:00Z
review text: This paper is concerned with the asymptotic behaviour of the zeros of polynomials that are extremal with respect to slowly decaying weights on \([0,\infty)\). A specific examples will indicate the nature of the work. Consider the log-normal weight function given by \(w^2(x)= \exp(-\gamma^2(\log x)^2)\), where \(\gamma> 0\). The corresponding extremal monic polynomial \(S_n(x, \gamma)\) of degree \(n\) satisfies the orthogonality condition \(\int^\infty_0 x^kS_n(x, \gamma) w^2(x) dx= 0\) for \(k= 0,\dots, n-1\), and its zeros \(x_{1,n},\dots, x_{n,n}\) are real and simple. Let \(\delta(x)\) denote the Dirac measure at \(x\). Then \(n^{-1} \sum^n_{j=1} \delta(x^{1/n}_{j,n})\) is shown to tend to a weak\(^*\) limit \(\rho\) as \(n\to\infty\). Further, the measure \(\rho\) is supported by the interval \([1,\exp(\gamma^{- 2})]\) and has density there given by \(d\rho/dx= \gamma^2 x^{-1}\). The arguments involve logarithmic potential theory.
Mathematics Subject Classification ID: 31A15
Mathematics Subject Classification ID: 33D45
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 1525173
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: zero distribution
zbMATH Keywords: asymptotic behaviour
zbMATH Keywords: logarithmic potential theory
author: Arno B. J. Kuijlaars
reviewed by: Stephen J. Gardiner
MaRDI profile type: Publication
full work available at URL: s003650010001
OpenAlex ID: W2013860425
Wikidata QID: Q127236620
DOI: 10.1007/S003650010001
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description: scientific article; zbMATH DE number 1525173