Entropy numbers of classes defined by integral operators

TL;DR

This paper introduces a novel approach to analyze the asymptotic behavior of entropy numbers for collections of function classes defined by integral operators with specific kernels, extending previous work on Kolmogorov widths.

Contribution

It develops a general method to study entropy numbers for classes defined via integral operators, broadening the scope beyond individual smoothness classes.

Findings

Established asymptotic estimates for entropy numbers of classes defined by integral operators.

Extended the analysis framework previously used for Kolmogorov widths.

Provided new insights into the complexity of function classes generated by integral kernels.

Abstract

In this paper we develop the following general approach. We study asymptotic behavior of the entropy numbers not for an individual smoothness class, how it is usually done, but for the collection of classes, which are defined by integral operators with kernels coming from a given class of functions. Earlier, such approach was realized for the Kolmogorov widths.