The structure of shift-invariant subspaces of Sobolev spaces

We analyze shift-invariant spaces \(V_s\) , subspaces of Sobolev spaces \(H^s(\mathbb{R}^n)\) , \(s\in\mathbb{R}\) , generated by a set of generators \(\varphi_i\) , \(i\in I\) , with \(I\) at most countable, by the use of range functions and characterize Bessel sequences, frames, and the Riesz basis of such spaces. We also describe \(V_s\) in terms of Gramians and their direct sum decompositions. We show that \(f\in\mathcal D_{L^2}'(\mathbb{R}^n)\) belongs to \(V_s\) if and only if its Fourier transform has the form \(\hat f=\sum_{i\in I}f_ig_i\) , \(f_i=\hat\varphi_i\in L_s^2(\mathbb{R}^n)\) , \(\{\varphi_i(\,\cdot+k)\colon k\in\mathbb Z^n,\,i\in I\}\) is a frame, and \(g_i=\sum_{k\in\mathbb{Z}^n}a_k^ie^{-2\pi\sqrt{-1}\,\langle\,{\cdot}\,,k\rangle}\) , with \((a^i_k)_{k\in\mathbb{Z}^n}\in\ell^2(\mathbb{Z}^n)\) . Moreover, connecting two different approaches to shift-invariant spaces \(V_s\) and \(\mathcal V^2_s\) , \(s>0\) , under the assumption that a finite number of generators belongs to \(H^s\cap L^2_s\) , we give the characterization of elements in \(V_s\) through the expansions with coefficients in \(\ell_s^2(\mathbb{Z}^n)\) . The corresponding assertion holds for the intersections of such spaces and their duals in the case where the generators are elements of \(\mathcal S(\mathbb R^n)\) . We then show that \(\bigcap_{s>0}V_s\) is the space consisting of functions whose Fourier transforms equal products of functions in \(\mathcal S(\mathbb R^n)\) and periodic smooth functions. The appropriate assertion is obtained for \(\bigcup_{s>0}V_{-s}\) .