Jackson theorems for Erdős weights in \(L_p\) \((0<p\leq \infty)\)
type: Item
instance of: scholarly article
title: Jackson theorems for Erdős weights in \(L_p\) \((0<p\leq \infty)\)
zbMATH Open document ID: 0915.41013
published in: Journal of Approximation Theory
publication date: 1999-02-14T00:00:00Z
review text: This paper is a the first part of a work on Jackson estimates for the approximation by polynomials, of functions in weighted \(L_p\), \(0<p\leq\infty\), where the weights are Erdős type weights. The second part contains inverse theorems and was written by the first author. For some reason the latter appeared earlier, in [J. Approximation Theory 93, No. 3, 349-398 (1998; Zbl 0907.41010]. In this paper the authors establish the Jackson estimates by introducing new moduli of smoothness, based on the Erdős weights and contain two parts. One which is given by the weighted norm of the differences of the function on a suitably defined finite interval, and the other which measures the best of approximation by polynomials of low degrees, on the infinite part of the interval. The above mentioned second paper shows that these moduli are indeed the proper ones. The proofs are quite technical. The authors first approximate the function by piecewise polynomials of low degree as best as possible (as guaranteed by Whitney), and then replace the piecewise polynomials by polynomials of the much higher degree.
reviewed by: D. Leviatan
Mathematics Subject Classification ID: 41A30
Mathematics Subject Classification ID: 41A10
zbMATH DE Number: 1216704
zbMATH Keywords: Jackson theorems
zbMATH Keywords: Erdős weights
zbMATH Keywords: approximation in \(L_p\)
author: S. B. Damelin
author: Doron S. Lubinsky
MaRDI profile type: Publication
full work available at URL: jath.1998.3199
OpenAlex ID: W1983668451
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DOI: 10.1006/JATH.1998.3199
description: scientific article; zbMATH DE number 1216704