Statistical \(\sigma \)-convergence of positive linear operators
type: Item
instance of: scholarly article
title: Statistical \(\sigma \)-convergence of positive linear operators
zbMATH Open document ID: 1223.40002
author: K. Demirci
author: Fadime Dirik
published in: Applied Mathematics Letters
publication date: 2011-01-18T00:00:00Z
review text: Denote by \(C_M[a,b]\) the space of all continuous functions on \([a,b]\) and bounded on the entire line, that is \(| f(x)|\leq K_f\), \(-\infty<x<\infty\), where \(K_f\) is a constant depending on \(f\). By \(B[a,b]\) is denoted the space of all bounded functions on \([a,b]\). In this paper, the author obtains the following Korovkin-type approximation theorem by means of the concept of statistical \(\sigma\)-convergence. Theorem. Let \(\{ L_m\}\) be a sequence of positive linear operators acting from \(C_M[a,b]\) into \(B[a,b]\). Assume that the following condition hold: \[ \delta(\sigma)-\lim\|L_m(f_i)-f_i\|_{B[a,b]}=0,\quad \text{where}\quad f_i(x)=x^i,\quad i=0,1,2. \] Then, for all \(f\in C_M[a,b]\), we have \[ \delta(\sigma)-\lim\|L_m(f)-f\|_{B[a,b]}=0. \] The rates of statistical \(\sigma\)-convergence of a sequence of positive linear operators are computed.
reviewed by: Ana Maria Acu
Mathematics Subject Classification ID: 40A35
Mathematics Subject Classification ID: 41A36
zbMATH DE Number: 5838267
zbMATH Keywords: statistical \(\sigma \)-convergence
zbMATH Keywords: statistical convergence
zbMATH Keywords: positive linear operator
zbMATH Keywords: Korovkin-type approximation theorem
zbMATH Keywords: Bernstein polynomials
MaRDI profile type: Publication
full work available at URL: j.aml.2010.10.031
OpenAlex ID: W2112596312
cites work: Q3272104
cites work: Korovkin-type approximation theory and its applications
cites work: Q4014382
cites work: A Baskakov type generalization of statistical Korovkin theory
cites work: Statistical fuzzy approximation by fuzzy positive linear operators
cites work: A-Statistical convergence of approximating operators
cites work: Statistical rates on the multivariate approximation theory
cites work: Statistical approximation of certain positive linear operators constructed by means of the Chan-Chyan-Srivastava polynomials
cites work: Some approximation theorems via statistical convergence.
cites work: Equi-statistical convergence of positive linear operators
cites work: On the invariant mean and statistical convergence
cites work: Sur la convergence statistique
cites work: ON STATISTICAL CONVERGENCE
cites work: Invariant means and invariant matrix methods of summability
DOI: 10.1016/J.AML.2010.10.031
description: scientific article; zbMATH DE number 5838267