Bernstein polynomials and modulus of continuity
type: Item
instance of: scholarly article
title: Bernstein polynomials and modulus of continuity
zbMATH Open document ID: 0957.41017
author: Zhongkai Li
published in: Journal of Approximation Theory
publication date: 2000-08-29T00:00:00Z
review text: The author describes several properties related to the first order modulus of continuity, which are preserved by the operator given by Bernstein polynomials. Let the function \(\omega(t)\) on \([0,1]\) be a modulus of continuity. By \(H^\omega\) we denote the class of continuous functions on \([0,1]\) satisfying the inequality \(\omega(f,t)\leq\omega(t)\). The author proves: (iii)* if \(\omega(t)\) is a modulus of continuity, then \(f\in H^\omega\) implies that for all \(n\geq 1\), \(B_n(f;x)\in H^{2\omega}\); if \(\omega(t)\) is concave (upper convex), then \(f\in H^\omega\) implies that for all \(n\geq 1\), \(B_n(f;x)\in H^\omega\). (iv) if \(\omega(t)\) is a modulus of continuity, then for each \(n\geq 1\), \(B_n(\omega;t)\) is also a modulus of continuity and \(B_n(\omega;t)\leq 2\omega(t)\); if \(\omega(t)\) is concave, then for each \(n\geq 1\), \(B_n(\omega;t)\) is a concave modulus of continuity and \(B_n(\omega;t) \leq\omega (t)\). (v) if \(f (x)\) is a non-negative function such that \(x^{-1}f(x)\) is non-increasing on \([0, 1]\), then for each \(n\geq 1\), \(x^{-1} B_n(f;x)\) is non-increasing also. The author gives some examples. Remark of reviewer. For generalization of (iii)* see \textit{G. A. Anastassiou}, and \textit{C. Cottin}, and \textit{H. H. Gonska} [Analysis 11, No. 1, 43-57 (1991; Zbl 0722.41021)] proved: \(\omega(B_nf; \delta)\leq 1\cdot \omega^* (f;\delta)\), \(\delta\geq 0\), here the constant 1 is best possible, \(\omega^* (f;\delta)\) is the least concave majorant of \(\omega(f; \delta)\) (we have \(\omega(f; \delta)\leq \omega^* (f;\delta)\leq 2\omega (f; \delta))\). If \(\omega(t)\) is a moduls of continuity and \(f\in H^\omega\), we have \(\omega(B_nf; \delta)\leq\omega^* (f;\delta)\leq 2\omega (f;\delta)\leq 2\omega (\delta)\), hence for all \(n\geq 1\) \(B_nf\in H^{2\omega}\). If \(\omega(t)\) is concave, then \(f\in H^\omega (\omega (f;t)\leq \omega(t))\) implies that \(\omega^* (f;t) \leq\omega (t)\) and \(\omega(B_nf; t)\leq 1\cdo…
Mathematics Subject Classification ID: 41A36
Mathematics Subject Classification ID: 41A17
zbMATH DE Number: 1417557
zbMATH Keywords: modulus of continuity
zbMATH Keywords: Bernstein polynomials
zbMATH Keywords: preservation
MaRDI profile type: Publication
full work available at URL: jath.1999.3374
OpenAlex ID: W1990553417
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DOI: 10.1006/JATH.1999.3374
description: scientific article; zbMATH DE number 1417557