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 $\bigcup_{s>0}V_{-s}$.