Statistical deferred weighted mathcal{B}-summability and its applications to associated approximation theorems

T Pradhan,S K Paikray,H Dutta,B B Jena

doi:10.1186/s13660-018-1650-x

Abstract

The notion of statistical weighted mathcal{B}-summability was introduced very recently (Kadak et al. in Appl. Math. Comput. 302:80–96, 2017). In the paper, we study the concept of statistical deferred weighted mathcal{B}-summability and deferred weighted mathcal{B}-statistical convergence and then establish an inclusion relation between them. In particular, based on our proposed methods, we establish a new Korovkin-type approximation theorem for the functions of two variables defined on a Banach space C_{B}(mathcal{D}) and then present an illustrative example to show that our result is a non-trivial extension of some traditional and statistical versions of Korovkin-type approximation theorems which were demonstrated in the earlier works. Furthermore, we establish another result for the rate of deferred weighted mathcal{B}-statistical convergence for the same set of functions via modulus of continuity. Finally, we consider a number of interesting special cases and illustrative examples in support of our findings of this paper.

Highlights

In the interpretation of sequence spaces, the well-established traditional convergence has got innumerable applications where the convergence of a sequence demands that almost all elements are to assure the convergence condition, that is, every element of the sequence is required to be in some neighborhood of the limit
In the past few decades, statistical convergence has been an energetic area of research due essentially to the aspect that it is broader than customary convergence, and such hypothesis is talked about in the investigation in the fields of Fourier analysis, functional analysis, number theory, and theory of approximation
Remark 3 Letn∈N be the sequence given in Example 2

Summary

Statistical deferred weighted B-summability

Here we introduce the notions of deferred weighted B -statistical convergence and statistical deferred weighted B -summability by using the deferred weighted regular matrices (methods). As a characterization of the deferred weighted regular methods, we present the following theorem. A sequence (xn ) is said to be deferred weighted B -statistically convergent to a number L if, for each > 0, we have (K ) = 0. We say that the sequence (xn ) is statistically deferred weighted B -summable to a number. We prove the following theorem which determines the inclusion relation between the deferred weighted B -statistical convergence and the statistical deferred weighted B summability. If a sequence (xn ) is deferred weighted B -statistically convergent to a number L, it is statistically deferred weighted B -summable to the same number L, but the converse is not true. Let (xn ) be the deferred weighted B -statistically convergent to L, we have uniformly in i, where.

A Korovkin-type theorem via statistical deferred weighted B-summability

Rate of deferred weighted B-statistical convergence

Concluding remarks and observations

Methods