A Certain Class of Equi-Statistical Convergence Based on $(p, q)$-integers via Deferred Nörlund Mean and Related Approximation Theorems

Abstract

The concept of equi-statistical convergence is more general than that of the well-established statistical uniform convergence. In this paper, we have introduced the idea of equi-statistical convergence, statistical point-wise convergence and statistical uniform convergence under the difference operator including $(p, q)$-integers via deferred Nörlund statistical convergence so as to build up a few inclusion relations between them. We have likewise presented the notion of the deferred weighted (Nörlund type) equi-statistical convergence (presumably new) in view of difference sequence of order $r$ based on $(p, q)$-integers to demonstrate a Korovkin type approximation theorem and proved that our theorem is a generalization (non-trivial) of some well-established Korovkin type approximation theorems which were demonstrated by earlier authors. Eventually, we set up various fascinating examples in connection with our definitions and results.