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 $\varrho_D$ serves as a degree of freedom. 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.

翻译：我们研究在统一规范中复制核心Hilbert 空间的多变量函数的恢复。 我们的主要利益是获取相应的抽样数字的精密估计值。 我们从嵌入 $_ (D),\ varrho_D) $ (美元) 的相关缩压单数的衰减结果, 乘以由第一个单函数所覆盖的子空间的 Christoffel 函数的精度值。 在这里, $\ varrho_ D$ (美元) 是一个自由度。 作为我们获得的具有一般光度重量的定期 Sobolev 类型空间抽样数字的接近最佳上限的应用程序。 这些应用可以用统一规范中相应的基准近似近值的近似值来约束, 从而允许有纯度界限。 通过应用最近引入的与 Weaver 的直径相关的子取样技术, 我们损失的大多是 $\ sqrt n, 有时甚至更少。 最后, 我们指出与相应的 Kolmogorov 数字的关系。