A generalization of Kantorovich operators for convex compact subsets

TL;DR

This paper introduces a generalized class of Kantorovich operators on convex compact sets, unifying and extending classical operators with new approximation and preservation properties.

Contribution

It develops a broad framework for Kantorovich operators depending on parameters, encompassing classical and recent generalizations, with detailed approximation and preservation analyses.

Findings

Operators unify classical and modern Kantorovich variants

Provided convergence rate estimates for the operators

Discussed preservation of Lipschitz and convex functions

Abstract

In this paper we introduce and study a new sequence of positive linear operators acting on function spaces defined on a convex compact subset. Their construction depends on a given Markov operator, a positive real number and a sequence of probability Borel measures. By considering special cases of these parameters for particular convex compact subsets we obtain the classical Kantorovich operators defined in the one-dimensional and multidimensional setting together with several of their wide-ranging generalizations scattered in the literature. We investigate the approximation properties of these operators by also providing several estimates of the rate of convergence. Finally, the preservation of Lipschitz-continuity as well as of convexity are discussed