Approximation results for a general class of Kantorovich type operators

TL;DR

This paper introduces a broad class of Kantorovich-type integral operators on locally compact groups, establishing their convergence properties across various function spaces and demonstrating their applicability to multiple operator classes.

Contribution

It develops a unified theoretical framework for Kantorovich operators, including convergence results and applications to discrete and integral operators in diverse function spaces.

Findings

Operators converge pointwise and uniformly

Applicable to sampling, convolution, and Mellin operators

Convergence results hold in various Orlicz spaces

Abstract

We introduce and study a family of integral operators in the Kantorovich sense for functions acting on locally compact topological groups. We obtain convergence results for the above operators with respect to the pointwise and uniform convergence and in the setting of Orlicz spaces with respect to the modular convergence. Moreover, we show how our theory applies to several classes of integral and discrete operators, as the sampling, convolution and Mellin type operators in the Kantorovich sense, thus obtaining a simultaneous approach for discrete and integral operators. Further, we derive our general convergence results for particular cases of Orlicz spaces, as $L^p-$spaces, interpolation spaces and exponential spaces. Finally we construct some concrete example of our operators and we show some graphical representations.