Korovkin Second Theorem via B-Statistical A-Summability

and Applied Analysis 3 f continuous on R. We know that C(R) is a Banach space with norm 󵄩󵄩󵄩f 󵄩󵄩󵄩∞ := sup x∈R 󵄨󵄨󵄨f (x) 󵄨󵄨󵄨 , f ∈ C (R) . (12) We denote by C 2π (R) the space of all 2π-periodic functions f ∈ C(R) which is a Banach space with 󵄩󵄩󵄩f 󵄩󵄩󵄩2π = sup t∈R 󵄨󵄨󵄨f (t) 󵄨󵄨󵄨 . (13) The classical Korovkin first and second theorems statewhatfollows [15, 16]: Theorem I. Let (T n ) be a sequence of positive linear operators from C[0, 1] into F[0, 1]. Then lim n ‖T n (f, x) − f(x)‖ ∞ = 0, for all f ∈ C[0, 1] if and only if lim n ‖T n (f i , x) − e i (x)‖ ∞ = 0, for i = 0, 1, 2, where e 0 (x) = 1, e 1 (x) = x, and e 2 (x) = x. Theorem II. Let (T n ) be a sequence of positive linear operators fromC 2π (R) into F(R). Then lim n ‖T n (f, x)−f(x)‖ ∞ = 0, for allf ∈ C 2π (R) if and only if lim n ‖T n (f i , x)−f i (x)‖ ∞ = 0, for i = 0, 1, 2, where f 0 (x) = 1, f 1 (x) = cosx, and f 2 (x) = sinx. We write L n (f; x) for L n (f(s); x), and we say that L is a positive operator if L(f; x) ≥ 0 for all f(x) ≥ 0. The following result was studied by Duman [17] which is A-statistical analogue of Theorem II. Theorem A. Let A = (a nk ) be a nonnegative regular matrix, and let (T k ) be a sequence of positive linear operators from C 2π (R) into C 2π (R). Then for all f ∈ C 2π (R)