Strong equivalences of approximation numbers and tractability of weighted anisotropic Sobolev embeddings

Abstract

In this paper, we study multivariate approximation defined over weighted anisotropic Sobolev spaces which depend on two sequences ${\bf a}=\{a_j\}_{j\geq1}$ and ${\bf b}=\{b_j\}_{j\geq1}$ of positive numbers. We obtain strong equivalences of the approximation numbers, and necessary and sufficient conditions on ${\bf a}$, ${\bf b}$ to achieve various notions of tractability of the weighted anisotropic Sobolev embeddings.