On simultaneous density order from shift invariant subspaces in Sobolev spaces

The notion of simultaneous approximation order (m, k) from shift-invariant subspaces in Sobolev spaces was introduced in the paper by Zhao (1995). Moreover, a characterization of those principal shift-invariant subspaces that provide simultaneous approximation order (m, k) was proved there. In this note, we prove another characterization using dilated by some adequate expansive linear maps of a shift-invariant subspace. In addition, we introduce the notion of simultaneous density order (m, k) and give necessary and sufficient conditions on a shift-invariant subspace to have a simultaneous density desired. To give our conditions, we shall explain the behavior on a neighborhood of the origin of the Fourier transform of the generators of a shift-invariant subspace. For this, we will use the classical notion of approximate continuity.