A certain class of deferred weighted statistical B-summability involving (p,q)-integers and analogous approximation theorems

Abstract

The preliminary idea of statistical weighted B-summability was introduced by Kadak et al. [27]. Subsequently, deferred weighted statistical B-summability has recently been studied by Pradhan et al. [38]. In this paper, we study statistical versions of deferred weighted B-summability as well as deferred weighted 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 prove an approximation theorem (Korovkin-type) for functions of two variables defined on a Banach space C B (D) and demonstrated that, our theorem effectively improves 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 estimate 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. 1. Introduction, Preliminaries and Motivation Let ω be the set of all real valued sequences and call any subspace of ω the sequence space. Let (x k) be a sequence with real or complex terms. Suppose ∞ is the class of all bounded linear sequence spaces and let c, c 0 be the respective classes for convergent and null sequences with real or complex terms. We have, x ∞ = sup k |x k | (k ∈ N), and we recall here that under this norm, the above mentioned spaces are all Banach spaces. The space of difference sequence was initially studied by Kızmaz [30] and then it was extended to the difference sequence of natural order r (r ∈ N 0 := {0} ∪ N) by defining λ(∆ r) = {x = (x k) : ∆ r (x) ∈ λ, λ ∈ (∞ , c 0 , c)} ;

A certain class of statistical convergence of martingale sequences and its applications to Korovkin-type approximation

“…with (X k , F k ) as already mentioned in above Example 2.1. Now, by using our proposed operators (19), we calculate the values of the functions 1, 2τ and 3τ 2 as follows: so (20) holds true. Now, by applying (20) and Theorem 3.2, the condition (21) also holds true.…”

“…Moreover, the introduction of statistical probability convergence has enhanced the glory of this development. For some recent research works in this direction, see [2], [3], [4], [6], [7], [9], [11], [16] and [20].…”

Statistical convergence of martingale difference sequence via deferred weighted mean and Korovkin-type theorems

“…This concept is extremely valuable in Real Analysis, Functional Analysis, Harmonic Analysis, and so on. Here, we like to refer the interested readers to the recent works [4,13,18,20,21] and [26].…”

Statistical product convergence of martingale sequences and its applications to Korovkin‐type approximation theorems

“…Moreover, the introduction of statistical probability convergence has enhanced the glory of this development. For some recent research works in this direction, see previous studies 3–15 …”