Korovkin type approximation theorems proved via αβ-statistical convergence

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 αβ-statistical convergence and αβ-statistical convergence of order γ. We show that αβ-statistical convergence is a non-trivial extension of ordinary and statistical convergences. Moreover we show that αβ-statistical convergence includes statistical convergence, λ-statistical convergence, and lacunary statistical convergence. We also introduce the concept of αβ-equistatistical convergence which is a non-trivial extension of equistatistical convergence. Moreover, we prove that αβ-equistatistical convergence lies between αβ-statistical pointwise convergence and αβ-statistical uniform convergence. Finally we prove Korovkin type approximation theorems via αβ-statistical uniform convergence of order γ and αβ-equistatistical convergence of order γ.