Approximation by the Bernstein-Durrmeyer operator on a simplex
type: Item
instance of: scholarly article
title: Approximation by the Bernstein-Durrmeyer operator on a simplex
zbMATH Open document ID: 1197.33011
author: Feng Dai
author: Wang Kunyang
author: Hong-Wei Huang
published in: Constructive Approximation
publication date: 2010-06-09T00:00:00Z
review text: The authors investigate the Jacobi-type Bernstein-Durrmeyer operator \(M_{n,k}\) on the simplex \(T^d\) of \(\mathbb{R}^d.\) They prove that for \(f\in L^p(W_k;T^d)\) with \(1<p<\infty\), \[ K_{2,\Phi}(f,n^{-1})_{k,p}\leq c\|f-M_{n,k}|_{k,p}\leq c'K_{2,\Phi}(f,n^{-1})_{k,p}+c'n^{-1}\|f\|_{k,p} \] holds, where \(W_k\) denotes the uasual Jacobi weight on \(T^d\), \(K_{2,\Phi}(f,n^{-1})_{k,p}\) and \(\|.\|_{k,p}\) denote the second-order Ditzian-Totik K-functional and the \(L^p\)-norm with respect to the weight \(W_k\) on \(T^d\), respectively, and the constants \(c\) and \(c'\) are independent of \(f\) and \(n\). This is a very good contribution in the approximation theory, all the more it confirms a conjecture of Berens et al.
reviewed by: Ridha Sfaxi
Mathematics Subject Classification ID: 33C50
Mathematics Subject Classification ID: 42B08
Mathematics Subject Classification ID: 42C10
zbMATH DE Number: 5719079
zbMATH Keywords: Bernstein-Durrmeyer operator
zbMATH Keywords: K-functionals
zbMATH Keywords: Ditzian-Totik moduli of smoothness
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: simplex
MaRDI profile type: Publication
full work available at URL: s00365-008-9030-2
OpenAlex ID: W2114655171
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DOI: 10.1007/S00365-008-9030-2
description: scientific article; zbMATH DE number 5719079