Weighted polynomial approximation on the real line
type: Item
instance of: scholarly article
title: Weighted polynomial approximation on the real line
zbMATH Open document ID: 0885.41011
author: András Kroó
author: Jószef Szabados
published in: Journal of Approximation Theory
publication date: 1998-04-20T00:00:00Z
review text: One of the most important subjects in the last decades has been related to polynomial approximation on the whole real line with weight \(w(x)= \exp[-Q(x)]\), where \(Q(x)\) has polynomial growth at infinity. There are two main problems studied in the present paper: asymptotics for the Markov factors and for the rate of best approximation of \(|x|\) and Jackson-type estimates for the degree of best approximation of some classes of functions. The authors introduce the rate of best weighted approximation \[ E_n^*(f,Q)= \inf_{p\in M_n}|w(x)(f(x)- p(x))| \] in order to proceed their studies. The paper is divided into two sections. First of them, ``Asymptotics for \(M_n(Q)\) and \(E_n^* (|x|,Q)\)'' is devoted to the following result: \(M_n(Q)\sim \frac{1} {E_n^*(|x|,Q)}\sim I_n(Q)\), where \(M_n(Q)\) are so-called Markov factors and \[ I_n(Q)= \int_1^{Q^{\{-1\}}(n)} \frac{Q(t)}{t^k} dt. \] More than that, the authors gave a new proof of the upper bound for \(M_n(Q)\), which is considerably simpler than previous results and also allows to substantially relax the restrictions imposed upon the weight \(Q\). The second section, ``Order of weighted approximation on the real line'', contains the investigations of the magnitude's order of \(E_n^*(f,Q)\) in order to provide some Jackson-type estimates and to discuss lower bounds as well.
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 41A25
zbMATH DE Number: 810857
zbMATH Keywords: polynomial approximation
zbMATH Keywords: best approximation
zbMATH Keywords: Jackson-type estimates
zbMATH Keywords: Markov factors
MaRDI profile type: Publication
full work available at URL: jath.1995.1107
OpenAlex ID: W2053297825
DOI: 10.1006/JATH.1995.1107
description: scientific article; zbMATH DE number 810857