The structure of shift-invariant subspaces of Sobolev spaces

We analyze shift-invariant spaces , subspaces of Sobolev spaces , , generated by a set of generators , , with at most countable, by the use of range functions and characterize Bessel sequences, frames, and the Riesz basis of such spaces. We also describe in terms of Gramians and their direct sum decompositions. We show that belongs to if and only if its Fourier transform has the form , , is a frame, and , with . Moreover, connecting two different approaches to shift-invariant spaces and , , under the assumption that a finite number of generators belongs to , we give the characterization of elements in through the expansions with coefficients in . The corresponding assertion holds for the intersections of such spaces and their duals in the case where the generators are elements of . We then show that is the space consisting of functions whose Fourier transforms equal products of functions in and periodic smooth functions. The appropriate assertion is obtained for .