A Note on Sampling Recovery of Multivariate Functions in the Uniform Norm
Kateryna Pozharska, Tino Ullrich
Abstract
We study the recovery of multivariate functions from reproducing kernel Hilbert spaces in the uniform norm. Surprisingly, a certain weighted least squares recovery operator which uses random samples from a tailored distribution leads to near-optimal results in several relevant situations. The results are stated in terms of the decay of related singular numbers of the compact embedding into $L_2(D)$ multiplied with the supremum of the Christoffel function of the subspace spanned by the first $m$ singular functions. As an application we obtain new recovery guarantees for Sobolev type spaces related to Jacobi type differential operators, on the one hand, and classical multivariate periodic Sobolev type spaces with general smoothness weight on the other hand. By applying a recently introduced subsampling technique related to Weaver's conjecture we mostly lose a $\sqrt{\log n}$ factor, compared to the optimal worst-case error, and sometimes even less.