On the rate of convergence of the Durrmeyer operator for functions of bounded variation
type: Item
instance of: scholarly article
title: On the rate of convergence of the Durrmeyer operator for functions of bounded variation
zbMATH Open document ID: 0666.41017
DOI: 10.1016/0021-9045(87)90033-5
author: Shunsheng Guo
published in: Journal of Approximation Theory
publication date: 1987-01-01T00:00:00Z
review text: Let f be a function defined on [0,1]. The Durrmeyer operator \(M_ n\) applied to f is determined in terms of \[ M_ n(f,x)=(n+1)\sum^{n}_{k=0}p_{nk}(x)\int^{1}_{0}f(t)p_{nk}(t)dt \] where \(p_{nk}(x)=\left( \begin{matrix} n\\ k\end{matrix} \right)x^ k(1- x)^{n-k}\). An estimate for the rate of convergence of the Durrmeyer operator is given for functions of bounded variation at each point of (0,1). To this end properties of the kernel \(p_{nk}(x)\) and some results of probability are used. The function \(f(t)=| t-x|\) \((0<x<1)\) taken as an example shows that the estimate is essentially the best possible.
Mathematics Subject Classification ID: 41A36
zbMATH DE Number: 4090099
zbMATH Keywords: Durrmeyer operator
zbMATH Keywords: estimate
zbMATH Keywords: rate of convergence
reviewed by: M.-B. A. Babaev
MaRDI profile type: Publication
full work available at URL: 0021-9045(87)90033-5
OpenAlex ID: W2056774975
cites work: Q3897459
cites work: On the rate of convergence of Bernstein polynomials of functions of bounded variation
cites work: Sur l'approximation de fonctions intégrables sur l'interval-0,1-ferme par des polynômes de Bernstein modifies
cites work: Q4183968
description: scientific article; zbMATH DE number 4090099