Korovkin type aproximation theorem through statistical lacunary summability

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 l, 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 statistical lacunary summability to improve the result of [Ann. Univ. Ferrara, 57(2) (2011) 373-381] by using the test functions 1, e(-x), e(-2x) in the place of 1, x and x(2). We apply the classical Baskakov operator to construct an example in support of our main result.