Weighted Convergence and Applications to Approximation Theorems for Sequences of Monotone and Sublinear Operators

In this paper, we utilize a convergence termed weighted convergence, which is expressed via weighted density. This extends asymptotic (also known in the literature as natural density and linear density) and logarithmic densities. We employ this to express and prove Korovkin-type theorems for sequences of operators that are monotone and sublinear (msLOs). To demonstrate the practical relevance of our results, illustrative examples that satisfy the conditions of our theorems are included. The rate of convergence for these msLOs is analysed using the modulus of continuity.