Summary: The concepts of equi-statistical convergence, statistical pointwise convergence and statistical uniform convergence for sequences of functions were introduced recently by M. Balcerzak
et al. [J. Math. Anal. Appl. 328, No. 1, 715–729 (2007; Zbl 1119.40002
)]. In this paper, we use the notion of
-statistical convergence in order to generalize these concepts. We establish some inclusion relations between them. We apply our new notion of
-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
-equi-statistical convergence. Some interesting examples are also displayed here in support of our definitions and results.