Shift-invariant subspaces of Sobolev type

This paper has the characteristics of a review paper in which results of shift-invariant subspaces of Sobolev type are summarized without proofs. The structure of shift-invariant spaces Vₛ, s, generated by at most countable family of generators, which are subspaces of Sobolev spaces Hˢ(Rⁿ), are announced in {aap} and Bessel sequences, frames and Riesz families of such spaces are characterized. With the Fourier multiplier (1-Δ/4π²)s/2f=F⁻¹((1+|t|²)s/2f(t)), we are able to extend notions and theorems in {MB} to spaces of the Sobolev type.