Statistical deferred weighted [Formula: see text]-summability and its applications to associated approximation theorems

• Published in: Journal of inequalities and applications Vol. 2018; no. 1; p. 65
• Main Authors: Pradhan, T, Paikray, S K, Jena, B B, Dutta, H
• Format: Journal Article
• Language: English
• Published: England 2018
• Subjects: Statistical deferred weighted [Formula: see text]-summability; Korovkin-type approximation theorems; Positive linear operators; Banach space; Rate of convergence; Deferred weighted [Formula: see text]-statistical convergence; Statistical convergence
• ISSN: 1025-5834
• DOI: 10.1186/s13660-018-1650-x

The notion of statistical weighted [Formula: see text]-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 [Formula: see text]-summability and deferred weighted [Formula: see text]-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 [Formula: see text] 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 [Formula: see text]-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.