Weighted Statistical Relative Invariant Mean in Modular Function Spaces with Related approximation Results. (18th August 2018)

• Record Type: Journal Article
• Authors: Kadak, Uğur
• Is Part Of: Numerical functional analysis and optimization. Volume 39:Number 11(2018)
• Page Start: 1181
• Page End: 1207
• Publication Date: 2018-08-18
• Subjects: Bivariate type of Stancu-Schurer-Kantorovich operators -- double weighted σ-density -- Korovkin-type approximation theorem by double sequences of positive linear operators -- relative modular weighted statistical σ-convergence -- relative modular weighted σ-statistical convergence; Primary 40G15, 41A35, 46E30 -- Secondary 46E80, 40B05, 40A30; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2018.1470096
• Languages: English
• ISSNs: 0163-0563

Abstract

Abstract: The idea of statistical relative convergence on modular spaces has been introduced by Orhan and Demirci. The notion of σ -statistical convergence was introduced by Mursaleen and Edely and further extended based on a fractional order difference operator by Kadak. The concern of this paper is to define two new summability methods for double sequences by combining the concepts of statistical relative convergence and σ -statistical convergence in modular spaces. Furthermore, we give some inclusion relations involving the newly proposed methods and present an illustrative example to show that our methods are nontrivial generalizations of the existing results in the literature. We also prove a Korovkin-type approximation theorem and estimate the rate of convergence by means of the modulus of continuity. Finally, using the bivariate type of Stancu-Schurer-Kantorovich operators, we display an example such that our approximation results are more powerful than the classical, statistical, and relative modular cases of Korovkin-type approximation theorems.