Bernstein-type approximation of set-valued functions in the symmetric difference metric

TL;DR

This paper extends Bernstein approximation techniques to set-valued functions using a new weighted set average, achieving a set-valued Weierstrass theorem and analyzing approximation rates in the symmetric difference metric.

Contribution

It introduces a novel weighted average of sets and adapts Bernstein operators for set-valued functions, establishing a set-valued Weierstrass theorem and approximation rate results.

Findings

New weighted average of sets with key properties

Set-valued Bernstein operators approximate H"older continuous SVFs

Asymptotic approximation rate matches real-valued functions

Abstract

We study the approximation of univariate and multivariate set-valued functions (SVFs) by the adaptation to SVFs of positive samples-based approximation operators for real-valued functions. To this end, we introduce a new weighted average of several sets and study its properties. The approximation results are obtained in the space of Lebesgue measurable sets with the symmetric difference metric. In particular, we apply the new average of sets to adapt to SVFs the classical Bernstein approximation operators, and obtain a set-valued analog of the Weierstrass approximation theorem. The rate of approximation of H\"older continuous SVFs by Bernstein operators is studied and shown to be asymptotically equal to that for real-valued functions. Finally, the results obtained in the metric space of sets are generalized to metric spaces endowed with an average satisfying certain properties.