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 Γj, j∈J, be a countable family of closed, co-compact
subgroups of a second countable locally compact abelian group G and study
systems of the form ∪j∈J{gj,p(⋅−γ)}γ∈Γj,p∈Pj with generators gj,p in L2(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 continuous variants, are GTI systems. Under a technical α
local integrability condition (α-LIC) we characterize when GTI systems
constitute tight and dual frames that yield reproducing formulas for L2(G).
This generalizes results on generalized shift invariant systems, where each
Pj is assumed to be countable and each Γj 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
Γj, our characterizations improve known results since the class of GTI
systems satisfying the α-LIC is strictly larger than the class of GTI
systems satisfying 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 L2(G). In
addition, we will see that the admissibility conditions for the continuous and
discrete wavelet and Gabor transform in L2(Rn) are special cases
of the same general characterizing equations.Comment: Minor changes (v2). To appear in Trans. Amer. Math. So