SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1363-1384, June 2022.
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.

中文翻译：

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统一范数下多元函数的抽样恢复注意事项

SIAM 数值分析杂志，第 60 卷，第 3 期，第 1363-1384 页，2022 年 6 月。
我们研究了通过在统一范数中再现核希尔伯特空间来恢复多元函数。令人惊讶的是，使用来自定制分布的随机样本的某个加权最小二乘恢复算子在几种相关情况下会导致接近最优的结果。结果表示为紧凑嵌入到 $L_2(D)$ 中的相关奇异数的衰减乘以由第一个 $m$ 奇异函数跨越的子空间的 Christoffel 函数的上确界。作为一个应用，我们一方面获得了与 Jacobi 类型微分算子相关的 Sobolev 类型空间的新恢复保证，另一方面获得了具有一般平滑权重的经典多元周期 Sobolev 类型空间的新恢复保证。通过应用最近引入的与 Weaver 相关的子采样技术