How anisotropic mixed smoothness affects the decay of singular numbers of Sobolev embeddings
How anisotropic mixed smoothness affects the decay of singular numbers of Sobolev embeddings
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DOI：
10.1016/j.jco.2020.101523
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发表时间：
2020-01
期刊：
影响因子：
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通讯作者：
T. Kuehn;W. Sickel;T. Ullrich
中科院分区：
文献类型：
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作者：
T. Kuehn;W. Sickel;T. Ullrich
我们继续研究高维Hilbert-Sobolev型张量积嵌入奇异数的渐近衰减性和预渐近衰减性，特别强调了基础维度d的影响。本文主要研究具有不同光滑性的一元Sobolev型空间的张量积。我们研究了在L 2和H1中的嵌入。换句话说，当函数只有n个线性度量时，我们考察了L 2和H1中的最坏情况的逼近误差。该领域的最新进展表明，对于仅使用函数值的恢复界来说，奇异数的精确界是必不可少的。我们设定的渐近界是很久以前就知道的。本文给出了n在预渐近范围内的正确渐近常数和显式界，补充和改进了文献中的几个结果。此外，我们改进了来自光滑度向量适度增加的设置的误差界，这已经由Papageorgiou和WoźNiakowski研究过。
We continue the research on the asymptotic and preasymptotic decay of singular numbers for tensor product Hilbert–Sobolev type embeddings in high dimensions with special emphasis on the influence of the underlying dimension d. The main focus in this paper lies on tensor products involving univariate Sobolev type spaces with different smoothness. We study the embeddings into L 2 and H 1. In other words, we investigate the worst-case approximation error measured in L 2 and H 1 when only n linear measurements of the function are available. Recent progress in the field shows that accurate bounds on the singular numbers are essential for recovery bounds using only function values. The asymptotic bounds in our setting are known for a long time. In this paper we contribute the correct asymptotic constant and explicit bounds in the preasymptotic range for n. We complement and improve on several results in the literature. In addition, we refine the error bounds coming from the setting where the smoothness vector is moderately increasing, which has been already studied by Papageorgiou and Woźniakowski.