Rate of convergence of Bernstein polynomials for functions with derivatives of bounded variation
type: Item
instance of: scholarly article
title: Rate of convergence of Bernstein polynomials for functions with derivatives of bounded variation
zbMATH Open document ID: 0688.41008
DOI: 10.1016/0022-247X(89)90211-4
published in: Journal of Mathematical Analysis and Applications
publication date: 1989-01-01T00:00:00Z
review text: Let \(S_ n(x)\) be the Bernstein polynomial of degree n for the function \(f_ x(t):=| t-x|\) evaluated at \(t=x\). The authors first show that \(S_ n(x)=(2x(1-x)/\pi n)^{1/2}+O(1/n)\) and use this to give the estimate \(| B_ n(f,x)-f(x)-\sigma (x(1-x)/2\pi n)^{1/2}| =O(1/n)\) for the approximation of functions f with derivative \(f'\) of bounded variation by Bernstein polynomials \(B_ n(f,x)\) of degree n. Here \(\sigma\) is the jump \(\sigma =f'(x+0)-f'(x-0)\) in the derivative at the position x. The authors also give expressions for the O(1/n) on the right-hand side in terms of the total variation of certain functions associated with \(f'\).
reviewed by: Rudolf Wegmann
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 41A17
zbMATH DE Number: 4127801
zbMATH Keywords: Bernstein polynomial
author: Ranko Bojanic
author: Fuhua Cheng
MaRDI profile type: Publication
cites work: Der Wert einiger Konstanten in der Theorie der Approximation mit Bernstein-Polynomen
cites work: The Bernstein Polynomials for Discontinuous Functions
cites work: On the rate of convergence of Bernstein polynomials of functions of bounded variation
cites work: Q5819609
description: scientific article; zbMATH DE number 4127801