Ideal relatively equal convergence involving difference operators with associated approximation theorems

Abstract

We introduce the notions of ideal relatively equal and uniform convergence inconjunction with difference operators of sequences of functions (${\mathcal{I}}(\Delta^{j}_{r,equi})$-convergence and ${\mathcal{I}}(\Delta^{j}_{r,u})$-convergence, respectively, for short). Under some condition, we obtain an equavalence relation by means of aforesaid notions. The Korovkin-type result is obtained through our newly notion of ${\mathcal{I}}(\Delta^{j}_{r,equi})$-convergence and constract an example by taking $\lambda$-Bernstein operators to support this result. Moreover, we analyze the rate of ${\mathcal{I}}(\Delta^{j}_{r,equi})$-convergence by utilizing the modulus of continuity.