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 $V_s$, $s\in \mathbb{R}$, generated by at most countable family of generators, which are subspaces of Sobolev spaces $H^s({\mathbb{R}}^n)$, are announced in \cite{aap} and Bessel sequences, frames and Riesz families of such spaces are characterized. With the Fourier multiplier ${(1-\frac{\mathrm{\Delta }}{4{\pi }^2})}^{s\mathrm{/}2}f={\mathcal{F}}^{-1}((1+\mathrm{\mid }t{\mathrm{\mid }}^2{)}^{s\mathrm{/}2}\widehat{f}(t))$, we are able to extend notions and theorems in \cite{MB} to spaces of the Sobolev type.