A characterization of tight and dual generalized translation invariant frames

We present results concerning generalized translation invariant (GTI) systems on a second countable locally compact abelian group G. These are systems with a family of generators {g
_{j, P}
}
_{jεJ, pεPJ}
⊂ L
²
(G), where J is a countable index set, and P
_j
, j ε J are certain measure spaces. Furthermore, for each j we let Γ
_j
, be a closed subgroup of G such that G/Γ
_j
is compact. A GTI system is then the collection of functions U
_jεJ
{g
_{j, p}
(· - γ}
_{γεΓj, pεPj}
. Many well known systems, such as wavelet, shearlet and Gabor systems, both the discrete and continuous types, are GTI systems. We characterize when such systems form tight frames, and when two GTI Bessel systems form dual frames for L
²
(G). In particular, this offers a unified approach to the theory of discrete and continuous frames and, e.g., yields well known results for discrete and continuous Gabor and wavelet systems.