On regular \(n\)th root asymptotic behavior of orthonormal polynomials
type: Item
instance of: scholarly article
title: On regular \(n\)th root asymptotic behavior of orthonormal polynomials
zbMATH Open document ID: 0728.42016
DOI: 10.1016/0021-9045(91)90117-S
published in: Journal of Approximation Theory
publication date: 1991-01-01T00:00:00Z
review text: Let \(P_ n(z)\), \(n=0,1,2,...\), be the orthonormal polynomials with respect to a positive measure \(\mu\) with compact support \(S(\mu)\subset {\mathbb{R}}.\) These polynomials are said to possess regular asymptotic behavior if \(\lim_{n\to \infty}| P_ n(z)|^{1/n}=e^{g(z,\infty)}\) locally uniformly on \({\mathbb{C}}\setminus [a,b],\) where \([a,b]\) is the smallest interval containing S(\(\mu\)) and \(g(z,\infty)\) is the (generalized) Green's function of the domain \({\mathbb{C}}\setminus S(\mu)\) with logarithmic pole at \(\infty\). Two important theorems are proved. The first theorem is a characterization theorem in which regular asymptotic behavior on a subset of the support S(\(\mu\)) is compared with the asymptotic behavior of other polynomials. Equivalences between different types of asymptotic behavior (on and off the support) are proved. The second theorem is a localization theorem in which the asymptotic behavior of the orthonormal polynomials is characterized by the asymptotic behavior of polynomials orthonormal with respect to restrictions of the measure. Extensive use is made of logarithmic potential theory.
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 31A15
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 4203014
zbMATH Keywords: orthonormal polynomials
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: Green's function
zbMATH Keywords: localization theorem
zbMATH Keywords: logarithmic potential theory
author: Herbert Stahl
reviewed by: Walter Van Assche
MaRDI profile type: Publication
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full work available at URL: 0021-9045(91)90117-s
OpenAlex ID: W2015029897
description: scientific article; zbMATH DE number 4203014