A Note on Sampling Recovery of Multivariate Functions in the Uniform Norm

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