Approximation of functions with small mixed smoothness in the uniform norm

TL;DR

This paper investigates the asymptotic behavior of multivariate function classes with small mixed smoothness in the uniform norm, focusing on approximation, entropy, and Kolmogorov numbers, with implications for sampling recovery.

Contribution

It provides new estimates for approximation and entropy numbers for functions with mixed smoothness up to 1/2, using convolution operators and interpolation techniques.

Findings

Bounds for best m-term trigonometric approximation

Decay rates of Kolmogorov and entropy numbers

Implications for sampling recovery in L2

Abstract

In this paper we present results on asymptotic characteristics of multivariate function classes in the uniform norm. Our main interest is the approximation of functions with mixed smoothness parameter not larger than $1/2$. Our focus will be on the behavior of the best $m$-term trigonometric approximation as well as the decay of Kolmogorov and entropy numbers in the uniform norm. It turns out that these quantities share a few fundamental abstract properties like their behavior under real interpolation, such that they can be treated simultaneously. We start with proving estimates on finite rank convolution operators with range in a step hyperbolic cross. These results imply bounds for the corresponding function space embeddings by a well-known decomposition technique. The decay of Kolmogorov numbers have direct implications for the problem of sampling recovery in $L_2$ in situations where recent results in the literature are not applicable since the corresponding approximation numbers are not square summable.