Journal of Complexity ( IF 1.7 ) Pub Date : 2020-10-13 , DOI: 10.1016/j.jco.2020.101523 Thomas Kühn , Winfried Sickel , Tino Ullrich
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 . The main focus in this paper lies on tensor products involving univariate Sobolev type spaces with different smoothness. We study the embeddings into and . In other words, we investigate the worst-case approximation error measured in and when only 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 . 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.
中文翻译:
各向异性混合光滑度如何影响Sobolev嵌入的奇异数的衰减
我们继续研究高维张量积Hilbert-Sobolev型嵌入的奇异数的渐近和渐近衰减,并特别强调基础维的影响 。本文的主要重点在于张量积,该张量积涉及具有不同平滑度的单变量Sobolev型空间。我们研究嵌入 和 。换句话说,我们研究了最坏情况下的近似误差 和 仅在 该功能可以进行线性测量。该领域的最新进展表明,单数上的准确界限对于仅使用函数值的恢复界限至关重要。在我们的环境中,渐近边界是众所周知的。在本文中,我们为以下问题贡献了正确的渐近常数和在渐近前范围内的显式界。我们补充并改进了文献中的一些结果。此外,我们细化了来自平滑度矢量适度增加的设置的误差范围,这已经由Papageorgiou和Woowskiniakowski进行了研究。