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

Abstract

Abstract The notion of statistical weighted B $\mathcal{B}$-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 B $\mathcal{B}$-summability and deferred weighted B $\mathcal{B}$-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 CB(D) $C_{B}(\mathcal{D})$ 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 B $\mathcal{B}$-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.

Keywords

• Statistical convergence
• Statistical deferred weighted B $\mathcal{B}$ -summability
• Deferred weighted B $\mathcal{B}$ -statistical convergence
• Positive linear operators
• Rate of convergence
• Banach space