Optimal recovery of linear operators from information of random functions

TL;DR

This paper develops optimal methods for recovering linear operators from noisy function information, focusing on derivatives in Sobolev spaces and solutions to the heat equation, with implications for stochastic error handling.

Contribution

It introduces new optimal recovery techniques that selectively utilize information, improving accuracy for derivatives and heat equation solutions under stochastic errors.

Findings

Optimal recovery methods for derivatives from Fourier transform data.

Optimal recovery of heat equation solutions from stochastic information.

Enhanced accuracy in operator recovery under noise conditions.

Abstract

The paper concerns problems of the recovery of linear operators defined on sets of functions from information of these functions given with stochastic errors. The constructed optimal recovery methods, in general, do not use all the available information. As a consequence, optimal methods are obtained for recovering derivatives of functions from Sobolev classes by the information of their Fourier transforms given with stochastic errors. A similar problem is considered for solutions of the heat equation.