Korovkin type approximation theorems proved via ??-statistical convergence

Abstract

The concept of statistical convergence was introduced by H. Fast, and studied by various authors. Recently, by using the idea of statistical convergence, M. Balcerzak, K. Dems and A. Komisarski introduced a new type of convergence for sequences of functions called equistatistical convergence. In the present paper we introduce the concepts of alpha beta-statistical convergence and alpha beta-statistical convergence of order gamma. We show that afi-statistical convergence is a non-trivial extension of ordinary and statistical convergences. Moreover we show that 43-statistical convergence includes statistical convergence, A-statistical convergence, and lacunary statistical convergence. We also introduce the concept of alpha beta-equistatistical convergence which is a non-trivial extension of equistatistical convergence. Moreover, we prove that alpha beta-equistatistical convergence lies between alpha beta-statistical pointwise convergence and afi-statistical uniform convergence. Finally we prove Korovkin type approximation theorems via up-statistical uniform convergence of order gamma and alpha beta-equistatistical convergence of order y. (C) 2013 Elsevier B.V. All rights reserved.