Let \(G\) be a second countable, locally compact abelian group and \( H\) a countable closed discrete subgroup with compact quotient \(G/H\). One defines a (principal) shift-invariant subspace \(V_\varphi=\overline{\mathrm{span}}\{\varphi(\cdot-k): k\in H\}\subset L^2(G)\).
The most general result of this work (Theorem 4.7) provides reconstruction in terms of shift-specific average samples. It does require that \(G\) is compactly generated, and thus isomorphic to \(\mathbb{R}^a\times\mathbb{Z}^b\times \mathbb{T}^c\times F_1\times F\) where \(F_1\) is finite and \(F\) is compact. It is assumed that \(a\) or \(c\) is nonzero (in particular, \(G\) is not discrete) and that there is a compact symmetric neighborhood \(\widetilde{N}\) of the identity of \(\widehat{G}\) such that \(|\sum_{j\neq h\in H} \bar{c}(h) c(j)\int_{\tilde{N}} \langle h-j,\gamma\rangle d\gamma|\leq \alpha\|c\|^2\) for all \(c\in \ell^2(H)\) (\(0<\alpha<|\widetilde{N}|\)). Here, \(\langle \cdot,\cdot\rangle\) is the dual pairing on \((G,\widehat{G})\). This assumption holds for \(G=\mathbb{R}^d\).
Suppose that \(\{\varphi(\cdot-h): h\in H\}\) forms a frame for \(V_\varphi\) and that there is a \(\beta>0\) such that \(\widehat\varphi(\gamma)>\beta\), \(\gamma\in \widetilde{N}\). Suppose also that \[ \max\{ \sup_{j\in H}\sum_{h\in H} |\varphi(h-x-j)-\varphi(h-j)|,\sup_{h\in H}\sum_{j\in H} |\varphi(h-x-j)-\varphi(h-j)|\}\leq L\, \] and that \(\sum_{h\in H} \sum_{j\in H} |\varphi(h-j)|^2 <\infty\). Then there is a frame \(\{S_j\}_{h\in H}\) of \(V_\varphi\) such that each \(f\in V_\varphi\) admits an expansion \[ f(x)=\sum_{h\in H} (f\ast \omega_h)(h) S_h(x),\quad x\in G\, . \]
A simpler, more concrete version (Theorem 4.4) that applies without the restriction on the structure of \(G\) is described as follows. Define \(E_\varphi=\{\gamma\in \widehat{G}: P_\varphi(\gamma)>0\}\) where \(P_\varphi(\gamma)=\sum_{\lambda\in H^\bot}|\widehat\varphi(\gamma+\lambda)|^2\) (\(\gamma\in \widehat{G}\)). A preliminary result (Theorem 2.4) states that for \(\varphi\in L^2(G)\), \(\{\varphi(\cdot-h)\}_{h\in H}\) forms a frame for \(V_\varphi\) if and only if there are constants \(0<A\leq B<\infty\) such that \(A\leq P_\varphi(\gamma)\leq B\) a.e. on \(E_\varphi\).
Assume this, and let \(\omega\in L^1\cap L^2 (G)\) be compactly supported. Theorem 4.4 states that the following are equivalent:

(i)

There exists a frame \(\{S(\cdot -h): h\in H\}\) for \(V_\varphi\) such that for all \(f\in V_\varphi\), \(f(x)=\sum_{h\in H} (f\ast\omega)(h) S(x-h)\) with convergence in \(L^2\) and in \(L^\infty\), and

(ii)

there exist \(0<A<B<\infty\) such that \(A\chi_{E_\varphi}\leq |\Psi_\omega|\leq B\chi_{E_\varphi}\) a.e., where \(\Psi_\omega(\gamma)=\sum_{h\in H} (\varphi\ast\omega)(h)\langle -h,\gamma\rangle\).

Several preliminary results are provided that tie sampling in with the structure of \(G\). Some of these are quantified in terms of the Zak transform of \(\varphi\) (with respect to \(H\)) defined by \(Z_\varphi(x,\gamma)=\sum_{h\in H} \varphi(x+h) \langle -h,\gamma\rangle\), \(x\in G\), \(\gamma\in \widehat{G}\). Just as in the case of \(L^2(\mathbb{R})\), the Zak transform satisfies a quasi-periodicity property that plays a role.