Weighted statistical convergence based on generalized difference operator involving (p,q)-gamma function and its applications to approximation theorems

Abstract

In this work, following a new approach of Baliarsingh (2016) [6], we introduce the concepts of statistically weighted ΨΔp,q-summability, weighted ΨΔp,q-statistical convergence and weighted strongly ΨΔp,q-summability with respect to the difference operator Δh,p,qα,β,γ including (p,q)-analogue of Gamma function. Some inclusion relations between newly proposed methods are examined. We then prove a Korovkin type approximation theorem for functions of two variables and also present an example via (p,q)-analogue of modified Bernstein–Schurer operators to show that our proposed method is stronger than its classical and weighted statistical versions. Furthermore, we compute the rate of convergence of approximating positive linear operators through the modulus of continuity. Finally, we present computational and geometrical approaches to illustrate some of our results in this paper.