A general approximation approach for the simultaneous treatment of integral and discrete operators

TL;DR

This paper introduces a unified approach to analyze the convergence of both discrete and integral operators on topological groups, applicable across various function spaces including Orlicz and L^p spaces.

Contribution

It presents a general framework for studying the convergence of a broad class of operators, encompassing many well-known cases, in a unified manner.

Findings

Established uniform convergence results for the operators.

Proved modular convergence in Orlicz spaces, covering many function space settings.

Unified treatment of integral and discrete operators on topological groups.

Abstract

In this paper we give a unitary approach for the simultaneous study of the convergence of discrete and integral operators described by means of a family of linear continuous functionals acting on functions defined on locally compact Hausdorff topological groups. The general family of operators introduced and studied includes very well-known operators in the literature. We give results of uniform convergence, and modular convergence in the general setting of Orlicz spaces.The latter result allow us to cover many other settings as the $L^p$-spaces, the interpolation spaces, the exponential spaces and many others.