Bounds for Lebesgue functions for Freud weights
type: Item
instance of: scholarly article
title: Bounds for Lebesgue functions for Freud weights
zbMATH Open document ID: 0819.41001
author: D. M. Matjila
published in: Journal of Approximation Theory
publication date: 1994-12-14T00:00:00Z
review text: The role of the Lebesgue function of Lagrange interpolation is well understood for arrays of interpolation points that lie in a fixed finite interval, for example, \([-1,1]\). The size of the Lebesgue function governs the pointwise convergence of the Lagrange interpolation process. It was G. Freud who initiated the study of the Lebesgue function of Lagrange interpolation for arrays of interpolation points over the whole real line. More precisely, he estimated the Lebesgue function when the interpolation points are chosen at the \(n\)th stage to be zeros of the Hermite polynomial of degree \(n\), orthogonal with respect to the weight \(W^ 2_ 2\), where \(W_ 2(x):= \exp(-{1\over 2} x^ 2)\). In this interesting paper, the author extends these results to large classes of Freud weights that include \(W_ \beta(x):= \exp(-{1\over 2}| x|^ \beta)\), \(\beta> 1\). He provides upper and lower bounds that are the same apart from a multiplicative constant. Even for the case of the Hermite weight, these improve on Freud's results. As a sample, if \(\{l_{jn}(x)\}^ n_{j= 1}\) denote the fundamental polynomials of Lagrange interpolation at the zeros \(\{x_{jn}\}^ n_{j= 1}\) of the orthogonal polynomial \(p_ n(x)\) of degree \(n\) with respect to the weight \(W^ 2_ \beta(x)\), then \[ W_ \beta(x) \sum^ n_{k= 1} | l_{kn}(x)|\sim 1+ n^{-1/(2\beta)}| p_ n W_ \beta| (x)\log n= O(\log n) \] uniformly for \(| x|\leq \sigma x_{1n}\). Here \(0< \sigma< 1\) is arbitrary and \(x_{1n}\) denotes the largest zero of \(p_ n\). Moreover, \(\sim\) means that the ratio of the two sides is bounded above and below by positive constants independent of \(n\).
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 42C99
zbMATH DE Number: 703288
zbMATH Keywords: Freud weights
zbMATH Keywords: Lagrange interpolation
zbMATH Keywords: Lebesgue function
reviewed by: Doron S. Lubinsky
MaRDI profile type: Publication
full work available at URL: jath.1994.1134
OpenAlex ID: W2074008819
DOI: 10.1006/JATH.1994.1134
description: scientific article; zbMATH DE number 703288