A new class of modified Bernstein operators
type: Item
instance of: scholarly article
title: A new class of modified Bernstein operators
zbMATH Open document ID: 0939.41016
author: Yasuo Kageyama
published in: Journal of Approximation Theory
publication date: 2000-07-03T00:00:00Z
review text: The Bernstein-Stancu operator \(B_{m,p}^{\langle\alpha ,\beta ,\gamma \rangle}f(x)\) [\textit{H. H. Gonska} and \textit{J. Meier}, Calcolo 21, 317-335 (1984; Zbl 0568.41021)] is used (for \(\beta = \gamma =p=0)\) to construct a new modified Bernstein operator \({}_{\alpha }B_n\) as the Maclaurin series truncated at degree \(\alpha \) of \(B_{m,0}^{\langle s,0,0\rangle}f(x)\) regarded as a function of \(s\) at the point \(s=-1/n.\) The main result is as follows. 1) For all \(\alpha ,p,q,r \in {\mathbb N}_0,\) there exists a constant \(M\) such that for all \(n\in {\mathbb N}\) and for all \(f\in C^r[0,1]\) \[ |(x-x^2)^p\bigl( {}_{\alpha }B_nf\bigr)^{(q+r)}(x)|\leq Mn^{q-\min \{p,[q/2]\}}\|f^{(r)}\|_{C[0,1]}; \] 2) For all \(\alpha ,\beta ,\gamma \in {\mathbb N}_0\) \((\beta \leq \alpha)\) and for all \(f\in C^{2\beta +\gamma }[0,1]\) \[ \|\bigl({}_{\alpha }B_n f\bigr)^{(\gamma)}-f{(\gamma)}\|=o(n^{-\beta }),n\to \infty; \] Moreover the explicit representations for the operator \({}_{\alpha }B_n\) and for the remainder \[ \lim_{n\to \infty }n^{\alpha +1}\bigl(({}_{\alpha }B_nf)^{(\gamma)}-f^{(\gamma)}\bigr) \] are given, and applications to numerical quadrature are discussed.
Mathematics Subject Classification ID: 41A35
zbMATH DE Number: 1389113
zbMATH Keywords: Bernstein-Stancu operator
zbMATH Keywords: Bernstein quasi-interpolants
reviewed by: Alexei Lukashov
MaRDI profile type: Publication
full work available at URL: jath.1999.3348
OpenAlex ID: W2091564217
cites work: Q3661894
cites work: Q5535148
cites work: Generalization of the left Bernstein quasi-interpolants
cites work: Q3934898
cites work: Q4195836
cites work: Q4207176
cites work: Q4022922
cites work: Q5552825
cites work: Q5579142
cites work: Q4002255
cites work: Norm of the Bernstein left quasi-interpolant operator
DOI: 10.1006/JATH.1999.3348
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description: scientific article; zbMATH DE number 1389113