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

• Published in: Journal of inequalities and applications 2018-03, Vol.2018 (1), p.1-21, Article 65
• Main Authors: Pradhan, T., Paikray, S. K., Jena, B. B., Dutta, H.
• Format: Article
• Language: English
• Subjects: Banach space; Deferred weighted B $\mathcal{B}$ -statistical convergence; Positive linear operators; Rate of convergence; Statistical convergence; Statistical deferred weighted B $\mathcal{B}$ -summability
• ISSN: 1029-242X; 1029-242X
• DOI: 10.1186/s13660-018-1650-x

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.