Steckin-Marchaud-type inequalities in connection with Bernstein polynomials
type: Item
instance of: scholarly article
title: Steckin-Marchaud-type inequalities in connection with Bernstein polynomials
zbMATH Open document ID: 0619.41013
DOI: 10.1007/BF01893435
published in: Constructive Approximation
publication date: 1986-01-01T00:00:00Z
review text: Let \(0\leq \alpha \leq 2\), \(\phi (x)=(x(1-x))^{1/2}\), \(C_ 0=\{f\in C[0,1]:\) \(f(0)=f(1)=0\}\), \(C_{\alpha}=\{f\in C_ 0:\phi^{-\alpha}f\) continuous and bounded on (0,1)\(\}\). For \(f\in C_{\alpha}\) let \(\| f\|_{\alpha}\) be the uniform norm of \(\phi^{-\alpha}f\). Let \(\Delta^*_ hf(x)=f(x+h)-2f(x)+f(x-h)\) and \[ \omega_{\alpha}(t,f)=\sup \{\phi^{-\alpha}(x)| \Delta^*_{h\phi (x)}f(x)|: x,x\pm h\phi (x)\in [0,1],\quad 0<h\leq t\}. \] Let \(B_ k\) be the classical Bernstein operators on C[0,1]. The author proves that for \(f\in C_{\alpha}\) and \(n\in N\) one has \(\omega_{\alpha}(n^{-},f)\leq M_{\alpha}n^{- 1}\sum^{n}_{k=1}\| B_ kf-f\|_{\alpha}.\) Using this Steckin- Marchaud-type inequality a number of known inverse theorems for Bernstein polynomials are derived and improved and a problem posed by V. Totik in 1984 is solved.
Mathematics Subject Classification ID: 41A27
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 41A36
zbMATH DE Number: 4004840
zbMATH Keywords: Bernstein operators
zbMATH Keywords: Steckin-Marchaud-type inequality
zbMATH Keywords: inverse theorems for Bernstein polynomials
author: Erich van Wickeren
reviewed by: Ioan Raşa
MaRDI profile type: Publication
cites work: An elementary proof of the inverse theorem for Bernstein polynomials
cites work: Q5677915
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cites work: An interpolation theorem and its applications to positive operators
full work available at URL: https://doi.org/10.1007/bf01893435
OpenAlex ID: W2027856607
description: scientific article; zbMATH DE number 4004840