Korovkin second theorem via statistical summability ( C , 1 )

Korovkin type approximation theorems are useful tools to check whether a given sequence (L n ) n1 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 -periodic functions on the real line. In this paper, we use the notion of statistical summability (C, 1) to prove the Korovkin approximation theorem for the functions 1, cos and sin in the space of all continuous 2 -periodic functions on the real line and show that our result is stronger. We also study the rate of weighted statistical convergence.