Generalized equi-statistical convergence of the deferred Nörlund summability and its applications to associated approximation theorems

Abstract

Statistical convergence has recently attracted the wide-spread attention of researchers due mainly to the fact that it is more general than the classical convergence. Furthermore, the notion of equi-statistical convergence is stronger than that of the statistical uniform convergence. Such concepts were introduced and studied by Balcerzak et al. (J Math Anal Appl 328:715–729, 2007). In this paper, we have used the notion of equi-statistical convergence, statistical point-wise convergence and statistical uniform convergence in conjunction with the deferred Norlund statistical convergence in order to establish several inclusion relations between them. We have also applied our presumably new concept of the deferred Norlund equi-statistical convergence to prove a Korovkin type approximation theorem and demonstrated that our theorem is a non-trivial extension of some well-known Korovkin type approximation theorems which were proven by earlier authors. Finally, we consider a number of interesting cases in support of our definitions and results presented in this paper.