In the present paper, we introduce a sequence of linear operators, which is a higher order generalization of positive linear operators defined by a class of Borel measures studied. Then, using the concept of A−statistical convergence we obtain some approximation results which are stronger than the aspects of the classical approximation theory. | Turk J Math 31 (2007) , 333 – 339. ¨ ITAK ˙ c TUB Higher Order Generalization of Positive Linear Operators Defined by a Class of Borel Measures Oktay Duman Abstract In the present paper, we introduce a sequence of linear operators, which is a higher order generalization of positive linear operators defined by a class of Borel measures studied in [2]. Then, using the concept of A−statistical convergence we obtain some approximation results which are stronger than the aspects of the classical approximation theory. Key Words: Statistical convergence, A-statistical convergence, positive linear operators, regular matrices, the elements of the Lipschitz class, Korovkin-type approximation theorem. 1. Introduction Let I be an arbitrary interval of the real line, and let C(I) denote the linear space of all real-valued continuous functions on I. Assume that g is a non-negative increasing function on [0, ∞) with g(0) = 1. If I is an unbounded interval, then we consider the following function space |f(y)| Cg (I) = f ∈ C(I) : lim = 0 for any c > 0 , (1) |y|→∞; (y∈I) (g(|y|))c which was examined in [2], [3]. Here, we should remark that if I = [a, +∞) (or I = (a, +∞)), then the item “|y| → ∞; (y ∈ I)” in the definition (1) reduces to “y → +∞”; however, if I = (−∞, a] (or I = (−∞, a)), then we have “y → −∞”. On the other hand, if I is an bounded interval, then we will use the space C(I) instead of Cg (I). 2000 Mathematics Subject Classification: 41A25, 41A36. 333 DUMAN Now, for each fixed x ∈ I, let {µn,x : n ∈ N} be a collection of measures defined on (I, B), where B is the sigma field of Borel measurable subsets of I. Assume that, for any δ > 0, the condition g(|y|)dµn,x(y) 0, lim ajn = 0. j n:|xn −L|≥ε We denote this limit by stA − lim x = L [7] (see also [12], [14]). If we take A = C1 , the Ces`aro matrix of order one, then C1 -statistical convergence is equivalent to statistical convergence [6], [8]. Also replacing the matrix A by the identity .
