Iterates of Bernstein operators, via contraction principle
type: Item
instance of: scholarly article
title: Iterates of Bernstein operators, via contraction principle
zbMATH Open document ID: 1056.41004
author: Ioan A. Rus
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-08-06T00:00:00Z
review text: Let \((B_n^m)_{m\in\mathbb N}\) denote the sequence of the iterates of the classical Bernstein operators \(B_n\). The author gives a simple proof of the following theorem, proved earlier, by \textit{R. P. Kelisky} and \textit{T. J. Rivlin} [Pac. J. Math. 21, 511--520 (1967; Zbl 0177.31302)]. Theorem. If \(n\in N^*\) is fixed, then, for all \(f\in C[0,1]\), \(\lim_{m\to\infty} B_n^m(f) (x)= f(0)+[f(1)-f(0)]x\), \(x \in[0,1]\).
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 41A36
zbMATH DE Number: 2081795
zbMATH Keywords: Bernstein operators
zbMATH Keywords: contraction principle
zbMATH Keywords: Picard operator
reviewed by: S. M. Mazhar
MaRDI profile type: Publication
full work available at URL: j.jmaa.2003.11.056
OpenAlex ID: W2068381316
cites work: On the iterates of some Bernstein-type operators
cites work: Iteration of positive approximation operators
cites work: Iterates of Bernstein polynomials
cites work: Generalized contractions and applications
cites work: Q4271322
DOI: 10.1016/J.JMAA.2003.11.056
description: scientific article; zbMATH DE number 2081795