Quantitative estimates in approximation by Bernstein-Durrmeyer-Choquet operators with respect to monotone and submodular set functions

TL;DR

This paper provides quantitative approximation estimates for generalized Bernstein-Durrmeyer operators involving Choquet integrals with respect to monotone and submodular set functions, extending classical results and offering practical applications.

Contribution

It introduces new quantitative error estimates for Bernstein-Durrmeyer operators using Choquet integrals, broadening approximation theory with applications to data analysis.

Findings

Derived bounds in terms of modulus of continuity and K-functional.

Established $L^p$-approximation results with error estimates.

Presented concrete examples improving classical error bounds.

Abstract

For the qualitative results of pointwise and uniform approximation obtained in \cite{Gal-Opris}, we present general quantitative estimates in terms of the modulus of continuity and in terms of a $K$-functional, for the generalized multivariate Bernstein-Durrmeyer operator $M_{n, \Gamma_{n, x}}$, written in terms of the Choquet integral with respect to a family of monotone and submodular set functions, $\Gamma_{n, x}$, on the standard $d$-dimensional simplex. When $\Gamma_{n, x}$ reduces to two elements, one a Choquet submodular set function and the other one a Borel measure, for suitable modified Bernstein-Durrmeyer operators, univariate $L^{p}$-approximations, $p\ge 1$, with estimates in terms of a $K$-functional are proved. In the particular cases when $d=1$ and the Choquet integral is taken with respect to some concrete possibility measures, the pointwise estimate in terms of the modulus of continuity is detailed. Some simple concrete examples of operators improving the classical error estimates are presented. Potential applications to practical methods dealing with data, like learning theory and regression models, also are mentioned.