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In this paper we connect the well established discrete frame theory ofgeneralized shift invariant systems to a continuous frame theory. To do so, welet $\Gamma_j$, $j \in J$, be a countable family of closed, co-compactsubgroups of a second countable locally compact abelian group $G$ and studysystems of the form $\cup_{j \in J}\{g_{j,p}(\cdot - \gamma)\}_{\gamma \in\Gamma_j, p \in P_j}$ with generators $g_{j,p}$ in $L^2(G)$ and with each $P_j$being a countable or an uncountable index set. We refer to systems of this formas generalized translation invariant (GTI) systems. Many of the familiartransforms, e.g., the wavelet, shearlet and Gabor transform, both theirdiscrete and continuous variants, are GTI systems. Under a technical $\alpha$local integrability condition ($\alpha$-LIC) we characterize when GTI systemsconstitute tight and dual frames that yield reproducing formulas for $L^2(G)$.This generalizes results on generalized shift invariant systems, where each$P_j$ is assumed to be countable and each $\Gamma_j$ is a uniform lattice in$G$, to the case of uncountably many generators and (not necessarily discrete)closed, co-compact subgroups. Furthermore, even in the case of uniform lattices$\Gamma_j$, our characterizations improve known results since the class of GTIsystems satisfying the $\alpha$-LIC is strictly larger than the class of GTIsystems satisfying the previously used local integrability condition. As anapplication of our characterization results, we obtain new characterizations oftranslation invariant continuous frames and Gabor frames for $L^2(G)$. Inaddition, we will see that the admissibility conditions for the continuous anddiscrete wavelet and Gabor transform in $L^2(\mathbb{R}^n)$ are special casesof the same general characterizing equations.

Reproducing formulas for generalized translation invariant systems on locally compact abelian groups