On optimal recovery in $L_2$

TL;DR

This paper establishes a bound on the optimal recovery error in the $L_2$ norm using Kolmogorov width in the uniform norm, providing a new tool for error estimation in function recovery.

Contribution

It proves a novel inequality linking $L_2$ recovery error to Kolmogorov width in the uniform norm, with applications to functions of mixed smoothness.

Findings

The optimal $L_2$ recovery error can be bounded by Kolmogorov width in the uniform norm.

The inequality is effective for classes of functions with mixed smoothness.

Provides a new approach for estimating errors in optimal function recovery.

Abstract

We prove that the optimal error of recovery in the $L_2$ norm of functions from a class $\bF$ can be bounded above by the value of the Kolmogorov width of $\bF$ in the uniform norm. We demonstrate on a number of examples of $\bF$ from classes of functions with mixed smoothness that the obtained inequality provides a powerful tool for estimating errors of optimal recovery.