各向异性Sobolev嵌入的预渐进行为和渐进行为
Preasymptotics and asymptotics of approximation numbers of anisotropic Sobolev embeddings
AI总结：
本文研究了各向异性Sobolev空间到L2空间的逼近数的预渐进行为和渐进行为，并证明了这些嵌入问题不可计算且不受到维度诅咒的影响。
摘要：
在本文中，我们获得了从各向异性Sobolev空间$W_2^{\bf R}(\Bbb T^d)$到$L_2(\Bbb T^d)$的嵌入的预渐进行为和渐进行为以及强等价性。我们还获得了从各向异性Sobolev空间$W_2^{\bf R}(\Bbb T^d)$的极限空间$W_2^{\infty}(\Bbb T^d)$到$L_2(\Bbb T^d)$的嵌入的预渐进行为。我们证明了上述两种嵌入问题都是不可计算的，并且不受到维度诅咒的影响。
英文摘要：
In this paper, we obtain the preasymptotic and asymptotic behavior and strong equivalences of the approximation numbers of the embeddings from the anisotropic Sobolev spaces $W_2^{\bf R}(\Bbb T^d)$ to $L_2(\Bbb T^d)$. We also get the preasymptotic behavior of the approximation numbers of the embeddings from the limit spaces $W_2^{\infty}(\Bbb T^d)$ of the anisotropic Sobolev spaces $W_2^{\bf R}(\Bbb T^d)$ to $L_2(\Bbb T^d)$. We show that both the above embedding problems are intractable and do not suffer from the curse of dimensionality.
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