A generalization of Kantorovich operators for convex compact subsets

Abstract

In this article, we introduce and study a new sequence of positive linear operators acting on function spaces dened on a convex compact subset.\ud Their construction depends on a given Markov operator, a positive real number, and a sequence of Borel probability measures. By considering special cases\ud of these parameters for particular convex compact subsets, we obtain the classical Kantorovich operators dened in the 1-dimensional and multidimensional\ud setting together with several of their wide-ranging generalizations scattered in the literature. We investigate the approximation properties of these operators\ud by also providing several estimates of the rate of convergence. Finally, we discuss the preservation of Lipschitz-continuity and of convexity

A sharpening of a problem on Bernstein polynomials and convex functions

“…If f ∈ C [0, 1] is convex the inequality Inequalities of type (1) have important applications. They are useful when studying whether the Bernstein-Schnabl operators preserve convexity (see [2,3,4]).…”

“…If (4) is valid, then (2) -and hence (1) -is a consequence of (3) and (4). Starting from these remarks, the second author presented the inequality (4) as an open problem in [10]. A probabilistic solution was found by A. Komisarski and T. Rajba [5] using the methods developed in [8] and [6].…”

A sharpening of a problem on Bernstein polynomials and convex functions and related results

“…During the Conference on Ulam's Type Stability (Rytro, Poland, 2014), Raşa [12] recalled his problem.Inequalities of type (1.1) have important applications. They are useful when studying whether the Bernstein-Schnabl operators preserve convexity (see [3,4]). Recently, J. Mrowiec, T. Rajba and S. Wąsowicz [10] affirmed the conjecture (1.1) in positive.…”

Multivariate weighted kantorovich operators

“…Recently, the classical Kantorovich operators received a lot of attention: see, e.g., [1], [2], [5], [6], [13]. In this paper we extend some results from [8] and [14], by introducing and studying multivariate weighted Kantorovich operators.…”