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The sequence x is statistically convergent to L provided that for each ε > 0, lim «~" 1 {the number of k < n: \x^ -L\ > ε} = 0.n In this paper we study a related concept of convergence in which the set {k: k < n) is replaced by {k: k r -\ < k < k r }, for some lacunary sequence {k r } . The resulting summability method is compared to statistical convergence and other summability methods, and questions of uniqueness of the limit value are considered.

Abstract. Following the concept of statistical convergence and statistical cluster points of a sequence x, we give a definition of statistical limit superior and inferior which yields natural relationships among these ideas: e.g., x is statistically convergent if and only if st-liminfx = st-limsupx. The statistical core of x is also introduced, for which an analogue of Knopp's Core Theorem is proved. Also, it is proved that a bounded sequence that is C 1 -summable to its statistical limit superior is statistically convergent.

Abstract. In this paper we provide various approximation results concerning the classical Korovkin theorem via A -statistical convergence. We also study the rates of A -statistical convergence of approximating positive linear operators and give some examples.

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Abstract. Using A-statistical convergence, we prove a Korovkin type approximation theorem which concerns the problem of approximating a function f by means of a sequence {T n (f ; x)} of positive linear operators acting from a weighted space C 1 into a weighted space B 2 .

In the present paper, we study a Kantorovich type generalization of Agratini's operators. Using A-statistical convergence, we will give the approximation properties of Agratini's operators and their Kantorovich type generalizations. We also give the rates of A-statistical convergence of these operators.

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