Weak limits of zeros of orthogonal polynomials
type: Item
instance of: scholarly article
title: Weak limits of zeros of orthogonal polynomials
zbMATH Open document ID: 0606.42022
DOI: 10.1007/BF01893436
author: J. L. Ullman
author: Matthew F. Wyneken
published in: Constructive Approximation
publication date: 1986-01-01T00:00:00Z
review text: Let \(P_ n(x,\mu)\) be the monic orthogonal polynomial associated with the positive unit Borel measure \(\mu\) with infinite support on \(I=[-1,1]\) and let \(\nu_ n=\nu_ n(\mu)\) denote the zero measure of \(P_ n(x,\mu)\), i.e., the unit measure with mass 1/n at each zero of \(P_ n(x,\mu)\). The sequence \(\{\nu_ n\}^{\infty}_{n=1}\) converges weakly to the measure \(\nu\) if \(\lim_{n\to \infty}\int fd\nu_ n=\int fd\nu\) for all functions f(x) continuous on I. A carrier is defined as a Borel subset of the support having unit measure and two measures are called carrier related if they share the same set of carriers. The authors demonstrate that the equilibrium measures are weak limits of the zero measures of the orthogonal polynomials associated with the carrier related measures.
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 30C15
Mathematics Subject Classification ID: 31C15
zbMATH DE Number: 3979731
zbMATH Keywords: Borel measure
zbMATH Keywords: carrier
zbMATH Keywords: equilibrium measures
zbMATH Keywords: weak limits
reviewed by: Luigi Gatteschi
MaRDI profile type: Publication
cites work: Q3333321
cites work: Q3259806
cites work: On the Regular Behaviour of Orthogonal Polynomials
cites work: Norm oscillatory weight measures
full work available at URL: https://doi.org/10.1007/bf01893436
OpenAlex ID: W2032319879
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description: scientific article; zbMATH DE number 3979731