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 (1-4²) ^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.