A certain class of weighted statistical convergence and associated Korovkin‐type approximation theorems involving trigonometric functions

• Published in: Mathematical methods in the applied sciences Vol. 41; no. 2; pp. 671 - 683
• Main Authors: Srivastava, H. M., Jena, Bidu Bhusan, Paikray, Susanta Kumar, Misra, U. K.
• Format: Journal Article
• Language: English
• Published: 30.01.2018
• Subjects: Banach space; deferred weighted statistical convergence; Korovkin‐type approximation theorems; periodic functions; positive linear operators; rate of convergence; statistical convergence
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.4636

The subject of statistical convergence has attracted a remarkably large number of researchers due mainly to the fact that it is more general than the well‐established theory of the ordinary (classical) convergence. In the year 2013, Edely et al introduced and studied the notion of weighted statistical convergence. In our present investigation, we make use of the (presumably new) notion of the deferred weighted statistical convergence to present Korovkin‐type approximation theorems associated with the periodic functions 1,cosx, and sinx defined on a Banach space C2π(R). In particular, we apply our concept of the deferred weighted statistical convergence with a view to proving a Korovkin‐type approximation theorem for periodic functions and also to demonstrate that our result is a nontrivial extension of several known Korovkin‐type approximation theorems which were given in earlier works. Moreover, we establish another result for the rate of the deferred weighted statistical convergence for the same set of functions. Finally, we consider a number of interesting special cases and illustrative examples in support of our definitions and of the results which are presented in this paper.