Statistical approximation by positive linear operators on modular spaces

Abstract

In this paper, we investigate the problem of statistical approximation to a function by means of positive linear operators defined on a modular space. Especially, in order to get more powerful results than the classical aspects we mainly use the concept of statistical convergence. A non-trivial application is also presented.

Abstract Korovkin-type theorems in modular spaces and applications

“…Also the case of not necessarily positive operators is considered, following an approach given in [5]. Our results extend Korovkin-type theorems given in [8,10,17,18] in the context of modular spaces and in [16] in the setting of ideal convergence. Note that at least the results concerning positive operators can be extended to more general kinds of convergence, not necessarily generated by free filters or regular matrix methods: among them we recall almost convergence [25].…”

“…In [10,17,18] some versions of the Korovkin theorem were given, with respect to methods of convergence, generated by a suitable non-negative regular summability matrix A. Note that for every such method there is a filter F with the property that the convergence generated by the matrix A is equivalent to the F-convergence, but the converse is in general not true [20, Lemma 4, Corollary 1].…”

Korovkin type approximation for double sequences via statistical A-summation process on modular spaces

“…Bardaro and Mantellini [4] introduced some Korovkin type approximation theorems via the notions of modular convergence and strong convergence. Afterwards Karakuş et al [11] investigated the modular Korovkin-type approximation theorem via statistical convergence and then, Orhan and Demirci [20] extended these type of approximations to the spaces of double sequences of positive linear operators as follows:…”

“…Another regular summability method introduced by Fast ( [8]) and which is not equivalent to any regular matrix method is called statistical convergence which is also known as (C, 1) statistical convergence. Furthermore, in recent years, various statistical approximation results and theorems have been proved via the concept of statistical convergence ( [9,11,19]) and the motivation using this type of convergence comes from that the obtained results are more powerful than the classical version of the approximations. One of these frequently used approximation method is the Korovkin-type approximation theorems.…”

Statistical Relative Approximation on Modular Spaces

“…Statistical convergence of single sequences was introduced by Steinhaus [24] and studied by many authors [11,12,14]. Recently, this concept was extended to the double sequences.…”