Approximation by Choquet Integral Operators

TL;DR

This paper introduces nonlinear approximation operators based on the Choquet integral, generalizing classical linear operators and demonstrating improved approximation properties for certain functions.

Contribution

It extends the Feller constructive scheme by incorporating nonlinear Choquet integral operators, including Bernstein-Choquet and Picard-Choquet, with proven approximation advantages.

Findings

Bernstein-Choquet and Picard-Choquet operators are introduced.

These operators show better approximation properties for some functions.

Qualitative and quantitative approximation results are established.

Abstract

The main aim of this paper is to show that the nonlinear Choquet integral can be used to construct nonlinear approximation operators, exactly as by the use in probability of the Lebesgue-type integral, linear and positive approximation operators are constructed. The so-called Feller constructive scheme is generalized, by introducing discrete and non-discrete nonlinear approximation operators in terms of the nonlinear Choquet integral with respect to a monotone and subadditive set function. As particular cases, Bernstein-Choquet and Picard-Choquet operators are introduced, for which qualitative and quantitative approximation properties are obtained. In some subclasses of functions, they have better approximation properties than the classical Berstein and Picard operators.