Ideal Relatively Uniform Convergence with Korovkin and Voronovskaya types Approximation Theorems

Abstract

We introduce the notion of ideally relative uniform convergence of sequences of real valued functions. We then apply this notion to prove Korovkin-type approximation theorem, and then construct an illustrative example by taking $(p,q)$-Bernstein operators which proves that our Korovkin theorem is stronger than its classical version as well as statistical relative uniform convergence. The rate of ideal relatively uniform convergence of positive linear operators by means of modulus of continuity is calculated. Finally, the Voronovskaya-type approximation theorem is also investigated.