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

In this paper we connect the well established discrete frame theory of generalized shift invariant systems to a continuous frame theory. To do so, we let j, j 2 J, be a countable family of closed, co-compact subgroups of a sec- ond countable locally compact abelian group G and study systems of the form (j2Jfgj;p( )g 2 j;p2Pj with generators gj;p in L 2 (G) and with each Pj being a countable or an uncountable index set. We refer to systems of this form as generalized translation invariant (GTI) systems. Many of the familiar transforms, e.g., the wavelet, shearlet and Gabor transform, both their discrete and continu- ous variants, are GTI systems. Under a technical local integrability condition ( -LIC) we characterize when GTI systems constitute tight and dual frames that yield reproducing formulas for L 2 (G). This generalizes results on generalized shift invariant systems, where each Pj is assumed to be countable and each j is a uni- form 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 j, our characterizations improve known results since the class of GTI systems satisfying the -LIC is strictly larger than the class of GTI systems sat- isfying the previously used local integrability condition. As an application of our characterization results, we obtain new characterizations of translation invariant continuous frames and Gabor frames for L 2 (G). In addition, we will see that the admissibility conditions for the continuous and discrete wavelet and Gabor trans- form in L 2 (R n ) are special cases of the same general characterizing equations.