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Generalized equi-statistical convergence of positive linear operators and associated approximation theorems. (English) Zbl 1255.41013
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 λ-statistical convergence in order to generalize these concepts. We establish some inclusion relations between them. We apply our new notion of λ-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 λ-equi-statistical convergence. Some interesting examples are also displayed here in support of our definitions and results.
41A36Approximation by positive operators