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

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, ̺ D ) multiplied with the supremum of the Christoffel function of the subspace spanned by the first m singular functions. Here the measure ̺ 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 √ log n and sometimes even less. Finally we point out a relation to the corresponding Kolmogorov numbers.

Universal discretization and sparse sampling recovery

“…with an absolute constant c > 0. We refer the reader for results on the sampling recovery in the uniform norm to the papers [29] and [15]. The reader can find a discussion of these results in [14], Section 2.5.…”

Sampling discretization and related problems

“…We refer the reader for some recent results on sampling recovery in the uniform norm to the paper [109].…”