A New Upper Bound for Sampling Numbers

Abstract We provide a new upper bound for sampling numbers $$(g_n)_{n\in \mathbb {N}}$$ ( g n ) ? N associated with the compact embedding of separable reproducing kernel Hilbert space into square integrable functions. There are universal constants $$C,c>0$$ C , c > 0 (which specified in paper) such that $$\begin{aligned} g^2_n \le \frac{C\log (n)}{n}\sum \limits _{k\ge \lfloor cn \rfloor } \sigma _k^2,\quad n\ge 2, \end{aligned}$$ 2 ? log ? k ? ? ? ? where $$(\sigma _k)_{k\in is sequence singular (approximation numbers) Hilbert–Schmidt $$\mathrm {Id}:H(K) \rightarrow L_2(D,\varrho _D)$$ Id : H K ? L D ? . The algorithm which realizes least squares based on specific set nodes. These constructed out random draw combination down-sampling procedure coming from celebrated proof Weaver’s conjecture, was shown to be equivalent Kadison–Singer problem. Our result non-constructive since we only show existence linear operator realizing above bound. general can instance applied well-known situation $$H^s_{\text {mix}}(\mathbb {T}^d)$$ mix s T d $$L_2(\mathbb $$s>1/2$$ 1 / obtain asymptotic g_n C_{s,d}n^{-s}\log (n)^{(d-1)s+1/2}, - + improves very recent results by shortening gap between and lower $$\sqrt{\log (n)}$$ implies dimensions $$d>2$$ any sparse grid recovery method does not perform asymptotically optimal.