Generalized equi-statistical convergence of positive linear operators and associated approximation theorems

The concepts of equi-statistical convergence, statistical pointwise convergence and statistical uniform convergence for sequences of functions were introduced recently by Balcerzak et al. [M. Balcerzak, K. Dems, A. Komisarski, Statistical convergence and ideal convergence for sequences of functions, J. Math. Anal. Appl. 328 (2007) 715–729]. In this paper, we use the notion of λλ-statistical convergence in order to generalize these concepts. We establish some inclusion relations between them. We apply our new notion of λλ-equi-statistical convergence to prove a Korovkin type approximation theorem and we show that our theorem is a non-trivial extension of some well-known Korovkin type approximation theorems. Finally, we prove a Voronovskaja type approximation theorem via the concept of λλ-equi-statistical convergence. Some interesting examples are also displayed here in support of our definitions and results.