Equi-statistical convergence of positive linear operators

Abstract

Balcerzak, Dems and Komisarski [M. Balcerzak, K. Dems, A. Komisarski, Statistical convergence and ideal convergence for sequences of functions, J. Math. Anal. Appl. 328 (2007) 715-729] have recently introduced the notion of equi-statistical convergence which is stronger than the statistical uniform convergence. In this paper we study its use in the Korovkin-type approximation theory. Then, we construct an example such that our new approximation result works but its classical and statistical cases do not work. We also compute the rates of equi-statistical convergence of sequences of positive linear operators. Furthermore, we obtain a Voronovskaya-type theorem in the equi-statistical sense for a sequence of positive linear operators constructed by means of the Bernstein polynomials.

Equi-statistical σ-convergence of positive linear operators

“…Later many researchers investigated these conditions for various operators defined on different spaces. Furthermore, in recent years, with the help of the concept of uniform statistical convergence, which is a regular (non-matrix) summability transformation, various statistical approximation results have been proved [8][9][10][11][12][13][14]. Then, it was demonstrated that those results are more powerful than the classical Korovkin theorem.…”

Korovkin type approximation theorems via lacunary equistatistical convergence

“…[2], [15], [17] and [23]. The concept of equistatistical convergence was introduced by Balcerzak et al [3] and was subsequently applied for establishing approximation theorems in [1], [10], [11] and [12]. In [1], Aktuglu and Gezer [1] generalized the idea of statistical convergence to lacunary equistatistical convergence.…”

A Voronovskaya-type formula for SMK operators via statistical convergence

“…In this case, a natural question arises: what should we do when the above limit fails? This question has been investigated in some recent papers [2,3,4,5,6,7,8,9,14,16] with the help of some convergence methods that are stronger than the usual convergence method, such as statistical convergence, A-statistical convergence, and ideal convergence.…”