Korovkin type approximation theorems for weighted $$\alpha \beta $$ α β -statistical convergence

Abstract

The concept of $$\alpha \beta $$ -statistical convergence was introduced and studied by Aktuğlu (Korovkin type approximation theorems proved via $$\alpha \beta $$ -statistical convergence, J Comput Appl Math 259:174–181, 2014). In this work, we generalize the concept of $$\alpha \beta $$ -statistical convergence and introduce the concept of weighted $$\alpha \beta $$ -statistical convergence of order $$\gamma $$ , weighted $$\alpha \beta $$ -summability of order $$\gamma $$ , and strongly weighted $$\alpha \beta $$ -summable sequences of order $$\gamma $$ . We also establish some inclusion relation, and some related results for these new summability methods. Furthermore, we prove Korovkin type approximation theorems through weighted $$\alpha \beta $$ -statistical convergence and apply the classical Bernstein operator to construct an example in support of our result.