A note on sampling recovery of multivariate functions in the uniform norm

TL;DR

This paper investigates the recovery of multivariate functions in the uniform norm from reproducing kernel Hilbert spaces, providing near optimal bounds for sampling numbers and exploring their relation to Kolmogorov numbers.

Contribution

It introduces preasymptotic estimates for sampling numbers using decay of singular numbers and Christoffel functions, with applications to Sobolev spaces and sub-sampling techniques.

Findings

Derived near optimal upper bounds for sampling numbers in Sobolev spaces.

Established bounds in terms of approximation numbers and Christoffel functions.

Linked sampling numbers to Kolmogorov numbers for multivariate function recovery.

Abstract

We study the recovery of multivariate functions from reproducing kernel Hilbert spaces in the uniform norm. Our main interest is to obtain preasymptotic estimates for the corresponding sampling numbers. We obtain results in terms of the decay of related singular numbers of the compact embedding into $L_2(D,\varrho_D)$ multiplied with the supremum of the Christoffel function of the subspace spanned by the first $m$ singular functions. Here the measure $\varrho_D$ is at our disposal. As an application we obtain near optimal upper bounds for the sampling numbers for periodic Sobolev type spaces with general smoothness weight. Those can be bounded in terms of the corresponding benchmark approximation number in the uniform norm, which allows for preasymptotic bounds. By applying a recently introduced sub-sampling technique related to Weaver's conjecture we mostly lose a $\sqrt{\log n}$ and sometimes even less. Finally we point out a relation to the corresponding Kolmogorov numbers.