Tractability of sampling recovery on unweighted function classes

TL;DR

This paper demonstrates that sampling recovery in unweighted Korobov and classical smoothness function spaces becomes computationally feasible when intersected with the Wiener algebra, leveraging compressed sensing techniques and non-linear algorithms.

Contribution

It shows that intersecting unweighted function classes with the Wiener algebra makes sampling recovery tractable, which was previously hindered by the curse of dimensionality.

Findings

Sampling recovery is tractable with non-linear algorithms.

Linear algorithms are insufficient for this problem.

The approach leverages compressed sensing theory.

Abstract

It is well-known that the problem of sampling recovery in the $L_2$-norm on unweighted Korobov spaces (Sobolev spaces with mixed smoothness) as well as classical smoothness classes such as H\"older classes suffers from the curse of dimensionality. We show that the problem is tractable for those classes if they are intersected with the Wiener algebra of functions with summable Fourier coefficients. In fact, this is a relatively simple implication of powerful results from the theory of compressed sensing. Tractability is achieved by the use of non-linear algorithms, while linear algorithms cannot do the job.