Preasymptotics and asymptotics of approximation numbers of anisotropic Sobolev embeddings
Provides theoretical insights into the complexity of approximating functions from anisotropic Sobolev spaces, relevant for high-dimensional numerical analysis.
The paper characterizes the preasymptotic and asymptotic behavior of approximation numbers for anisotropic Sobolev embeddings, proving that these problems are intractable but do not suffer from the curse of dimensionality.
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.