Triangular \(A\)-statistical approximation by double sequences of positive linear operators
type: Item
instance of: scholarly article
title: Triangular \(A\)-statistical approximation by double sequences of positive linear operators
zbMATH Open document ID: 1338.40009
author: Carlo Bardaro
author: A. Boccuto
author: K. Demirci
author: Ilaria Mantellini
author: Sevda Orhan
published in: Results in Mathematics
publication date: 2015-11-06T00:00:00Z
review text: The authors study a variation of \(A\)-statistical convergence of double sequences, called triangular \(A\)-statistical approximation, where, for a suitable matrix \(A=(a_{i,j})\), a double sequence \((x_{i,j})\) is triangular \(A\)-statistically convergent to \(L\) if, for every \(\varepsilon>0\), \[ \lim_{i \to \infty} \sum_{j \in K_i(\varepsilon)}a_{i,j}= 0, \] where \(K_i(\varepsilon) = \{ j\in \mathbb N : j \leq i,\, |x_{i,j}-L|\geq \varepsilon\}\). Examples are provided that illustrate that neither \(A\)-statistical convergence nor triangular \(A\)-statistical convergence imply the other. The authors also prove a Korovkin-type theorem on triangular \(A\)-statistical convergence. They compare this theorem to Korovkin-type theorems related to Pringsheim convergence and \(A\)-statistical convergence of double sequences, giving examples where each theorem can be applied, while the other two theorems can not be applied. The rates of triangular \(A\)-statistical convergence are also studied, and theorems are proven giving approximation properties for summability matrices in terms of these rates of convergence.
reviewed by: Jeremy Wade
Mathematics Subject Classification ID: 40A35
Mathematics Subject Classification ID: 41A35
Mathematics Subject Classification ID: 41A36
Mathematics Subject Classification ID: 40B05
zbMATH DE Number: 6505274
zbMATH Keywords: \(A\)-statistical convergence
zbMATH Keywords: triangular \(A\)-statistical convergence
zbMATH Keywords: triangular density
zbMATH Keywords: Korovkin theorem
zbMATH Keywords: rate of convergence
MaRDI profile type: Publication
full work available at URL: s00025-015-0433-7
OpenAlex ID: W2152811669
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DOI: 10.1007/S00025-015-0433-7
description: scientific article; zbMATH DE number 6505274