On relative weighted summability in modular function spaces and associated approximation theorems

Abstract

The purpose of the present article is to extend the notion of relative convergence on modular function spaces using the weighted double \((\alpha ,\beta )\)-density. We give the definitions of statistically relatively modularly \(\mathcal {P}\)-summability and its strong form for double sequences of functions. Also we derive some important inclusion relations concerning newly proposed methods and present an illustrative example to show that our method is a non-trivial generalization of the relatively modularly convergence. As an application, we prove a Korovkin type approximation theorem through the double sequences of positive linear operators defined on a modular function space. Moreover using bivariate case of Kantorovich-type generalization of the Bernstein–Schurer positive linear operators, we give some corollaries to demonstrate that our methods are stronger than their classical and statistical versions.