Sampling recovery on classes defined by integral operators and sparse approximation with adaptive dictionaries

TL;DR

This paper investigates the asymptotic behavior of sampling recovery errors across classes defined by integral operators, linking it to sparse approximation with adaptive dictionaries, advancing understanding in approximation theory.

Contribution

It extends the analysis of sampling recovery errors to classes defined by integral operators and connects this to adaptive sparse approximation methods.

Findings

Established asymptotic error behavior for classes defined by integral operators.

Linked sampling recovery errors to sparse nonlinear approximation with adaptive dictionaries.

Extended previous results from Kolmogorov widths and entropy numbers to broader classes.

Abstract

In this paper we continue to develop the following general approach. We study asymptotic behavior of the errors of sampling recovery 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 and very recently for the entropy numbers. It turns out that the above problem is closely related to the sparse approximation problem with respect to different redundant dictionaries. Specifically, the problem of sampling recovery is connected with sparse nonlinear approximation with respect to adaptive dictionaries, which means that the dictionary depends on the function under approximation.