A - Statistical extension of the Korovkin type approximation theorem

Let A = (a jn) be an infinite summability matrix. For a given sequence x := (xn), the A-transform of x, denoted by Ax := ((Ax) j), is given by (Ax) j = ∑n=1 a jnxn provided the series converges for each j ∈ N, the set of all natural numbers. We say that A is regular if limAx = L whenever limx = L [4]. Assume that A is a non-negative regular summability matrix. Then x = (xn) is said to be A-statistically convergent to L if, for every ε > 0, lim j ∑n:|xn−L|≥ε a jn = 0, which is denoted by stA − limx = L [9] (see also [13,16]). We note that by taking A = C1, the Cesàro matrix, A-statistical convergence reduces to the concept of statistical convergence (see [8,10,17] for details). If A is the identity matrix, then A-statistical convergence coincides with the ordinary convergence. It is not hard to see that every convergent sequence is A-statistically convergent. However, Kolk [13] showed that A-statistical convergence is stronger than convergence when A = (a jn) is a regular summability matrix such that lim j maxn |a jn|= 0. It should be noted that A-statistical convergence may also be given in normed spaces [14]. Approximation theory has important applications in the theory of polynomial approximation, in various areas of functional analysis [1,3,5,12,15]. The study of the Korovkin type approximation theory is a well-established area of research, which deals with the problem of approximating a function f by means of a sequence {Ln f} of positive linear operators. Statistical convergence, which was introduced nearly fifty years ago, has only recently become an area of active research. Especially it has made an appearance in approximation theory (see, for instance, [7,11]). The aim of the present paper is to investigate their use in approximation theory settings. Throughout this paper I := [0,∞). As usual, let C(I) := { f : f is a real-valued continuous functions on I}, and CB(I) := { f ∈ C(I): f is bounded on I}. Consider the space Hw of