Banach Journal of Mathematical Analysis

A generalization of Kantorovich operators for convex compact subsets

Francesco Altomare, Mirella Cappelletti Montano, Vita Leonessa, and Ioan Raşa

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In this article, 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 Borel probability measures. By considering special cases of these parameters for particular convex compact subsets, we obtain the classical Kantorovich operators defined in the 1-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, we discuss the preservation of Lipschitz-continuity and of convexity.

Article information

Banach J. Math. Anal., Volume 11, Number 3 (2017), 591-614.

Received: 16 July 2016
Accepted: 19 September 2016
First available in Project Euclid: 6 May 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47B65: Positive operators and order-bounded operators
Secondary: 41A36: Approximation by positive operators

Markov operator positive approximation process Kantorovich operator preservation property


Altomare, Francesco; Cappelletti Montano, Mirella; Leonessa, Vita; Raşa, Ioan. A generalization of Kantorovich operators for convex compact subsets. Banach J. Math. Anal. 11 (2017), no. 3, 591--614. doi:10.1215/17358787-2017-0008.

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