Preasymptotics and asymptotics of approximation numbers of anisotropic Sobolev embeddings

In this paper, we obtain preasymptotic and asymptotic behavior and strong equivalences of the approximation numbers of the embeddings from the anisotropic Sobolev spaces W2R(Td) to L2(Td). We also get the preasymptotic behavior of the approximation numbers of the embeddings from the limit spaces W2∞(Td) of the anisotropic Sobolev spaces W2R(Td) to L2(Td). We show that both the above embedding problems are intractable and do not suffer from the curse of dimensionality.