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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.

In this paper we consider power series method which is also member of the class of all continuous summability methods. The power series method includes Abel method as well as Borel method. We investigate, using the power series method, Korovkin type approximation theorems for the sequence of positive linear operators defined on C[a, b] and L q [a, b], 1 ≤ q < ∞, respectively. We also study some quantitative estimates for L q approximation and give the rate of convergence of these operators.

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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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Establishing a one-to-one correspondence between the interval (0, 1] and the collection of all subsequences of a given sequence s n , Buck and Pollard proved that s n is (C , 1)-summable if almost all of subsequences are, but not conversely. In this article, we consider the analogous questions for the p-Cesàro matrices. We show, for example, that if p > 1 2 then a bounded sequence is C p -summable if and only if almost all of its subsequences are C p -summable.