On a generalization of Bernstein-Chlodovsky operators for convex cones

Year 2025, Volume: 8 Issue: Special Issue: ICCMA, 1 - 17, 16.12.2025

Francesco Altomare

https://doi.org/10.33205/cma.1744040

Abstract

A sequence of positive linear operators acting on suitable function spaces on convex Borel cones is introduced and studied. Such operators generalize the well-known Bernstein-Chlodovsky operators on $[0,+\infty[$ and, in addition, they unify many of their more recent extensions to other settings. The study is mainly addressed to highlight their pointwise convergence as well as their uniform convergence on compact subsets for particular classes of bounded Borel measurable functions. In some particular cases, by means of such operators, a Weierstrass-type density result is obtained which concerns the approximation of such class of functions in terms of polynomials or, more generally, of elements of some function algebras. In order to achieve the main results some new Korovkin-type theorems are also discussed in the framework of completely regular spaces. In a final section, some examples and applications are discussed as well.

Keywords

Positive linear operator , Korovkin-type theorem , Bernstein-Schnabl operator , Bernstein-Chlodovsky operator , Weierstrass-type theorem , convex cone

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