关于一致范数下多元函数采样恢复的一个注记
A note on sampling recovery of multivariate functions in the uniform norm
AI总结：
本文研究再生核 Hilbert 空间中多元函数在一致范数下的采样恢复，利用奇异数、Christoffel 函数及与 Weaver 猜想相关的子采样技术，建立预渐近且近最优的采样数界。
摘要：
我们研究在一致范数下从再生核 Hilbert 空间中恢复多元函数。我们的主要兴趣是获得相应采样数的预渐近估计。我们根据紧嵌入到 $L_2(D,\varrho_D)$ 的相关奇异数的衰减，乘以前 $m$ 个奇异函数张成的子空间的 Christoffel 函数的上确界，得到了若干结果。这里测度 $\varrho_D$ 可由我们自行选择。作为应用，我们获得了具有一般光滑性权重的周期 Sobolev 型空间采样数的近最优上界。这些上界可以用一致范数下相应的基准逼近数来界定，从而允许给出预渐近界。通过应用最近提出的与 Weaver 猜想相关的子采样技术，我们主要损失一个 $\sqrt{\log n}$ 因子，有时甚至更少。最后，我们指出了与相应 Kolmogorov 数之间的关系。
英文摘要：
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 $\varrho_D$ is at our disposal. 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.
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