Approximation for periodic functions via weighted statistical convergence

Korovkin type approximation theorems are useful tools to check whether a given sequence (L-n)(n >= 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 weighted statistical convergence to prove the Korovkin approximation theorem for the functions 1, cos and sin in the space of all continuous 2 pi-periodic functions on the real line and show that our result is stronger. We also study the rate of weighted statistical convergence. (C) 2013 Elsevier Inc. All rights reserved.