A note on Bernstein's theorems
type: Item
instance of: scholarly article
title: A note on Bernstein's theorems
zbMATH Open document ID: 0831.41010
author: Johan Lithner
author: Adam P. Wójcik
published in: Journal of Approximation Theory
publication date: 1996-02-11T00:00:00Z
review text: The authors repeat an argument of \textit{W. Pawłucki} and \textit{W. Pleśniak} [Math. Ann. 275, 467-480 (1986; Zbl 0591.32016)] and \textit{W. Pleśniak} [J. Approximation Theory 61, No. 1, 106-117 (1990; Zbl 0702.41023)] (case of \({\mathcal C}^\infty\) functions) to show that, for a compact subset \(E\) of \(\mathbb{R}^N\) preserving Markov's inequality with exponent \(r\), if the distance of \(f:\to \mathbb{R}\) from the polynomials on \(\mathbb{R}^N\) at degree at most \(n\) is of order \(n^{-r (k+n)}\) with some \(k\in \mathbb{Z}_+\) and \(\rho\in (0,1)\), then \(f\) extends to a function \(\widetilde {f}\) from a Zygmund-type class \(\Lambda_{k+ \rho} (\mathbb{R}^N)\). An analogical observation for Lipschitz-type classes has already been published by the second author [ibid. 67, No. 3, 252-269 (1991; Zbl 0752.41035)].
reviewed by: Wiesław Pleśniak
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 41A65
zbMATH DE Number: 803678
zbMATH Keywords: Markov inequality
zbMATH Keywords: Lipschitz-type classes
zbMATH Keywords: Zygmund-type class
MaRDI profile type: Publication
full work available at URL: jath.1995.1053
OpenAlex ID: W1999657776
DOI: 10.1006/JATH.1995.1053
description: scientific article; zbMATH DE number 803678