Abstract Approximation properties of shift–invariant subspaces and their dilations in Sobolev spaces are studied. Indeed, we characterize those shift–invariant subspaces that provide a fixed simultaneous approximation order and/or simultaneous density order. Here, we work in a multidimensional context and consider dilations of shift–invariant subspaces by powers of a fix expansive linear map. We put emphasis on the finitely generated shift–invariant subspaces. To give our results on simultaneous density order we need the notion of approximate continuity associated to the considered expansive linear map. Since the linear maps can be aniisotropic, our conditions depend of such dilations.
Boukeffous et al. (Thu,) studied this question.