Preasymptotics and asymptotics of approximation numbers of anisotropic Sobolev embeddings

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 $W_2^{\mathbf{R}}({\mathbb{T}}^d)$ to $L_2({\mathbb{T}}^d)$. We also get the preasymptotic behavior of the approximation numbers of the embeddings from the limit spaces $W_2^{\mathrm{\infty }}({\mathbb{T}}^d)$ of the anisotropic Sobolev spaces $W_2^{\mathbf{R}}({\mathbb{T}}^d)$ to $L_2({\mathbb{T}}^d)$. We show that both the above embedding problems are intractable and do not suffer from the curse of dimensionality.