On Optimal Recovery of Integrals of Set-Valued Functions

TL;DR

This paper investigates optimal methods for approximating integrals of set-valued functions, considering both exact and noisy data, and aims to identify the best strategies based on the functions' continuity properties.

Contribution

It introduces a framework for optimal recovery of set-valued integrals using fixed or free sampling points, accounting for errors and continuity constraints.

Findings

Derived optimal recovery strategies for set-valued integrals.

Analyzed the impact of measurement errors on approximation quality.

Provided bounds for approximation errors based on function continuity.

Abstract

In this paper we consider the problem of optimization of approximate integration of set-valued functions from the class defined by given majorant of their moduli of continuity, using values of the functions at $n$ fixed or free points of their domain. We consider the cases of exact information and information with error.