论文标题
各向异性混合平滑度如何影响sobolev嵌入的奇异衰减
How anisotropic mixed smoothness affects the decay of singular numbers of Sobolev embeddings
论文作者
论文摘要
我们继续对张量产品Hilbert-Sobolev型嵌入的单数数字的渐近和预响衰变的研究，并特别强调了基础尺寸$ d $的影响。本文的主要重点在于张量产品涉及具有不同平滑度的单变量Sobolev类型的空间。我们将嵌入到$ l_2 $和$ h^1 $中。换句话说，我们研究了仅$ n $线性样本可用时，以$ l_2 $和$ h^1 $测量的最坏情况近似错误。该领域的最新进展表明，单数数字上的准确界限对于仅使用函数值的恢复边界至关重要。我们环境中的渐近界限很长一段时间以来。在本文中，我们以$ n $的贡献范围内贡献正确的渐近常数和明确的界限。我们在文献中补充并改善了一些结果。此外，我们完善了来自平滑度矢量中等增加的设置的误差界限，Papageorgiou和wo {c {g} niakowski已经对此进行了研究。
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 samples 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 Wo{ź}niakowski.
