Polynomial approximation and interpolation on the real line with respect to general classes of weights
type: Item
instance of: scholarly article
title: Polynomial approximation and interpolation on the real line with respect to general classes of weights
zbMATH Open document ID: 0919.41004
DOI: 10.1007/BF03322043
author: K. Appert
published in: Results in Mathematics
publication date: 1999-08-26T00:00:00Z
review text: In this interesting paper, the authors extend results on weighted polynomial approximation and Lagrange interpolation for Freud weights such as \(\exp(-| x| ^\alpha)\), \(\alpha>1\) by allowing the weight to vanish at finitely many points on the real axis. Recall Bernstein's approximation problem: let \(w:\mathbb{R}\to (0,\infty)\) be continuous and let \(C_w\) denote the space of continuous functions \(f:\mathbb{R}\to\mathbb{R}\) with \[ \lim_{| x|\to\infty} (fw)(x)=0 \] and with norm \[ \| f\|:= \sup_{x\in\mathbb{R}} | fw|(x). \] When are the polynomials dense in \(C_w\)? A necessary condition is \[ \int_{-\infty}^\infty \frac{|\log w(x)|} {1+x^2} dx=\infty. \] When \(w\) is even and \(|\log w(e^x)| \) is convex in \([0,\infty)\), this is necessary and sufficient. In particular for \(w(x)= \exp(-| x|^\alpha)\), the polynomials are dense if \(\alpha\geq 1\). The authors discuss the case where \(w\) vanishes (even as strongly as an essential singularity) at finitely many points on the real line. Then, as their main result, they prove that there is an array of Lagrange interpolation points giving a Lebesgue constant of order \(\log n\). This generalizes earlier work of the second author for Freud weights, where \(\log n\) is known to be optimal.
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 41A10
zbMATH DE Number: 1209935
zbMATH Keywords: Lebesgue constants
zbMATH Keywords: weighted polynomial approximation
zbMATH Keywords: Lagrange interpolation
zbMATH Keywords: Freud weights
reviewed by: Doron S. Lubinsky
MaRDI profile type: Publication
cites work: Functions of the second kind for Freud weights and series expansions of Hilbert transforms
cites work: $L_\infty $ Markov and Bernstein Inequalities for Freud Weights
cites work: Christoffel functions, orthogonal polynomials, and Nevai's conjecture for Freud weights
cites work: Q5688975
cites work: Strong asymptotics for extremal polynomials associated with weights on \({\mathbb{R}}\)
cites work: Where does the sup norm of a weighted polynomial live? (A generalization of incomplete polynomials)
cites work: Weighted Lagrange and Hermite-Fejér interpolation on the real line
description: scientific article; zbMATH DE number 1209935