A counterexample to subexponential growth of orthogonal polynomials
type: Item
instance of: scholarly article
title: A counterexample to subexponential growth of orthogonal polynomials
zbMATH Open document ID: 0852.42013
DOI: 10.1007/BF01208561
published in: Constructive Approximation
publication date: 1996-12-01T00:00:00Z
review text: Two systems of orthogonal polynomials on the real line are constructed. These orthogonal polynomials satisfy a three-term recurrence relation for which most of the recurrence coefficients are constant (as for Chebyshev polynomials), except for a thin subsequence. It is shown that \([- 1, 1]\) is in the support of the orthogonality measure but that the support contains points outside \([- 1, 1]\). The author then shows that at the endpoints of the support one has \(\liminf|p_n(x)|^{1/n}> 1\). This is quite interesting since it is known that \(\limsup|p_n(x)|^{1/n}\leq 1\) uniformly for \(x\) on the support of the orthogonality measure when the recurrence coefficients are asymptotically periodic. Szwarc's examples thus show that this subexponential growth of orthogonal polynomials on the support of the measure does not hold in general for measures on a compact support and that some asymptotic control of the recurrence coefficients indeed is needed.
reviewed by: Walter Van Assche
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 47B39
zbMATH DE Number: 810116
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: three-term recurrence relation
zbMATH Keywords: orthogonality measure
zbMATH Keywords: subexponential growth
zbMATH Keywords: support
author: Ryszard Szwarc
MaRDI profile type: Publication
Wikidata QID: Q124845213
cites work: Q4170993
cites work: On the asymptotic distribution of eigenvalues of banded matrices
cites work: Orthogonal polynomials: Their growth relative to their sums
cites work: A lower bound for orthogonal polynomials with an application to polynomial hypergroups
cites work: Chain sequences and compact perturbations of orthogonal polynomials
cites work: Relative growth of linear iterations and orthogonal polynomials on several intervals
full work available at URL: https://doi.org/10.1007/bf01208561
OpenAlex ID: W2133184941
description: scientific article; zbMATH DE number 810116