Generalized Bernstein-Chlodowsky polynomials
type: Item
instance of: scholarly article
title: Generalized Bernstein-Chlodowsky polynomials
zbMATH Open document ID: 0927.41018
author: R. O. Efendiev
author: Ertan Ibikli
author: Akif D. Gadjiev
published in: Rocky Mountain Journal of Mathematics
publication date: 1999-12-13T00:00:00Z
full work available at URL: CONT28-4.html
review text: For given positive integers \(n\) and \(m\), the generalization of Bernstein-Chlodowsky polynomials is defined by \[ B_{n,m}(f,x)= \Biggl( 1+(m-1) \frac{x}{b_n} \Biggr) \sum_{k=0}^{[n/m]} f\Biggl( \frac{b_nk} {n-(m-1)k}\Biggr) C_{n-(m-1)k}^k \Biggl( \frac{x}{b_n} \Biggr)^k \Biggl(1- \frac{x}{b_n} \Biggr)^{n-mk}, \] where \(b_n\) is a sequence of positive numbers such that \(\lim_{n\to\infty} b_n= \infty\), \(\lim_{n\to\infty} (b_n/n)=0\), \(0\leq x\leq b_n\) and \([p]\) denotes the greatest integer less than \(p\). Using a Korovkin type theorem [see \textit{A. D. Gadzhiev}, Math. Zametki 20, 781-786 (1976; Zbl 0383.41016)] the authors obtained a theorem about convergence of \(B_{n,m}(f,x)\) to \(f(x)\) as \(n\to\infty\) in the weighted space of functions \(f\) continuous on the positive semiaxis and satisfying the condition \(\lim_{x\to\infty} (f(x)/ (1+x^2))= k_f<\infty\). But the theorem of convergence rate is not established.
Mathematics Subject Classification ID: 41A63
Mathematics Subject Classification ID: 41A10
zbMATH DE Number: 1309878
zbMATH Keywords: Bernstein-Chlodowsky polynomials
MaRDI profile type: Publication
OpenAlex ID: W1970724204
cites work: Q3878114
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cites work: A note on a Bernstein-type operator of Bleimann, Butzer, and Hahn
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DOI: 10.1216/RMJM/1181071716
description: scientific article; zbMATH DE number 1309878