Generalization of equi-statistical convergence via weighted lacunary sequence with associated Korovkin and Voronovskaya type approximation theorems

We introduce the notions of weighted lacunary statistical pointwise and uniform convergence and a kind of convergence which is lying between aforementioned convergence methods, namely, weighted lacunary equi-statistical convergence and obtain various implication results with supporting examples. We then apply our new concept of weighted lacunary equi-statistical convergence with a view to proving Korovkin and Voronovskaya type approximation theorems. We also construct an example with the help of generating functions type Meyer-König and Zeller which shows that our Korovkin-type theorem is stronger than its classical version. Moreover, we compute the rate of weighted lacunary equi-statistical convergence for operators in terms of modulus of continuity.