Convergence in ψ-Density: Fundamental Properties and Approximation Theorems for Positive Linear Operators

We define a new convergence concept called convergence in ψ-density. Let D be the class of strictly increasing, differentiable, and unbounded functions ψ:0,∞⟶0,∞. We say that a sequence x=xk is convergent in ψ-density to L if the limit limi⟶∞1/∑k=1iψ′k∑k≤i,k∈Bψ′k=0, where B≔Bϵ≔k≤i:xk−L≥ϵ for every ϵ>0. If the function ψ∈D is concave, our convergence method strictly implies asymptotic density convergence, which is widely recognized as statistical convergence. In this paper, we first establish the fundamental properties of ψ-density convergence. Subsequently, we prove a Korovkin-type approximation theorem for sequences of positive linear operators (pLOs) under this new framework. We provide an illustrative example, supported by graphical representations, to verify our theoretical findings. Finally, we compute the rate of convergence of these operators in terms of the modulus of continuity.