A Certain Class of Deferred Weighted Statistical B-summability Involving (p, q)-integers and Analogous Approximation Theorems

Abstract

The preliminary idea of statistical weighted $\mathcal{B}$-summability was introduced by Kadak et al. \cite{25}. Subsequently deferred weighted statistical $\mathcal{B}$-summability has recently been studied by Pradhan et al. \cite{45}. In this paper, we have studied statistical versions of deferred weighted $\mathcal{B}$-summability as well as deferred weighted $\mathcal{B}$-convergence with respect to the difference sequence of order $r (>0)$ involving $(p,q)$-integers and accordingly
established an inclusion between them. Moreover, based upon our
proposed methods, we have proved an approximation theorem
(Korovkin-type) for function of two variables defined on a Banach
space $C_{B}(\mathcal{D})$ and demonstrated that, our theorem
effectively improve and generalizes most (if not all) of the existing
results depending on the choice of $(p, q)$-integers. Finally, with the help of the modulus of continuity we have estimated the rate of convergence for our proposed methods. Also, an illustrative example is provided here by generalized $(p, q)$-analogue of Bernstein operators of
two variables to demonstrate that our theorem is stronger than its traditional and statistical versions.