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Mursaleen, M.; Kiliçman, A.

Korovkin second theorem via \(B\)-statistical \(A\)-summability. (English) Zbl 1477.41012

Abstr. Appl. Anal. 2013, Article ID 598963, 6 p. (2013).
Summary: Korovkin type approximation theorems are useful tools to check whether a given sequence \((L_n)_{n \geq 1}\) of positive linear operators on \(C[0, 1]\) of all continuous functions on the real interval \([0, 1]\) is an approximation process. That is, these theorems exhibit a variety of test functions which assure that the approximation property holds on the whole space if it holds for them. Such a property was discovered by Korovkin in 1953 for the functions \(1, x\), and \(x^2\) in the space \(C[0, 1]\) as well as for the functions \(1, \cos\), and \(\sin\) in the space of all continuous \(2\pi\)-periodic functions on the real line. In this paper, we use the notion of \(B\)-statistical \(A\)-summability to prove the Korovkin second approximation theorem. We also study the rate of \(B\)-statistical \(A\)-summability of a sequence of positive linear operators defined from \(C_{2\pi}(\mathbb R)\) into \(C_{2\pi}(\mathbb R)\).

Cited in 12 Documents

MSC:

41A36 Approximation by positive operators
40A35 Ideal and statistical convergence
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