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首页> 外文期刊>Journal of complexity >How anisotropic mixed smoothness affects the decay of singular numbers for Sobolev embeddings
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How anisotropic mixed smoothness affects the decay of singular numbers for Sobolev embeddings

机译：各向异性的混合光滑度如何影响SoboLev Embeddings的奇异数字的衰减

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摘要

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 Wozniakowski. (C) 2020 Published by Elsevier Inc.
机译：我们继续研究张量产品Hilbert-SoboLev型孤立数的渐近数和贫微衰退的研究，以高尺寸的高尺寸，特别强调底层尺寸D的影响。本文的主要重点在于涉及单变量的SoboLev型空间具有不同光滑性的张量产品。我们将嵌入物研究到L-2和H-1中。换句话说，我们研究了L-2和H-1中测量的最坏情况近似误差，当仅具有N个功能的N线性测量值时。该领域的最新进展表明，仅使用函数值的奇数数对奇数数对恢复界限是必不可少的。我们设置中的渐近界限很长一段时间。在本文中，我们在否的缺乏症范围内贡献了正确的渐近常数和显式范围。我们在文献中补充并改进了几个结果。此外，我们可以通过Papageorgiou和Wozniakowski研究，优化来自平滑度向量的误差范围。 （c）2020由elsevier公司发布

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