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

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 $\mathbf{a}$, $\mathbf{b}$ to achieve various notions of tractability of the weighted anisotropic Sobolev embeddings.