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We study the concepts of I-limit and I-cluster points of a sequence, where I is an ideal with the Baire property. We obtain the relationship between I-limit and I-cluster points of a subsequence of a given sequence and the set of its classical limit points in the sense of category theory.

Abstract. In this paper, we present some relationships between convergence and uniform statistical convergence of a given sequence and its subsequences. The results concerning uniform statistical convergence presented here are also closely related to earlier results regarding statistical convergence and almost convergence of sequences, and are dealing with measure and in a minor case with category. Finally, we present a Cauchy type characterization of uniform statistical convergence and a result concerning uniform statistical convergence of subseries of a series.

In this paper, we present some results linking the uniform statistical limit superior and inferior, almost convergence and uniform statistical convergence of a sequence. We also study the relationship between the set of uniform statistical cluster points of a given sequence and its subsequences. The results concerning uniform statistical convergence and uniform statistical cluster points presented here are also closely related to earlier results regarding statistical convergence and statistical cluster points of a sequence.

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In the present study we introduce uniform statistical convergence for double sequences. We present a decomposition theorem that characterizes uniform statistical convergence for double sequences.

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In this paper we consider power series method which is also member of the class of all continuous summability methods. We study a Korovkin type approximation theorem for a sequence of positive linear operators acting from a weighted space C ρ 1 into a weighted space B ρ 2 with the use of the power series method which includes both Abel and Borel methods. We also consider the rates of convergence of these operators.