Korovkin-type theorems for abstract modular convergence

TL;DR

This paper extends Korovkin-type theorems to various modes of abstract modular convergence, providing convergence criteria, rate estimates, and applications to Mellin convolution and Kantorovich operators.

Contribution

It introduces new Korovkin-type theorems applicable to filter, ideal, almost, and A-statistical convergence, with practical applications in integral operators.

Findings

Established convergence theorems for nets of functions under various abstract convergence modes.

Provided estimates for rates of approximation in these convergence settings.

Applied the theoretical results to Mellin convolution and bivariate Kantorovich operators.

Abstract

We give some Korovkin-type theorems on convergence and estimates of rates of approximations of nets of functions, satisfying suitable axioms, whose particular cases are filter/ideal convergence, almost convergence and triangular A-statistical convergence, where A is a non-negative summability method. Furthermore, we give some applications to Mellin-type convolution and bivariate Kantorovich-type discrete operators.