In this work we extend classical structure and duality results in Gabor
analysis on the euclidean space to the setting of second countable locally
compact abelian (LCA) groups. We formulate the concept of rationally
oversampling of Gabor systems in an LCA group and prove corresponding
characterization results via the Zak transform. From these results we derive
non-existence results for critically sampled continuous Gabor frames. We obtain
general characterizations in time and in frequency domain of when two Gabor
generators yield dual frames. Moreover, we prove the Walnut and Janssen
representation of the Gabor frame operator and consider the Wexler-Raz
biorthogonality relations for dual generators. Finally, we prove the duality
principle for Gabor frames. Unlike most duality results on Gabor systems, we do
not rely on the fact that the translation and modulation groups are discrete
and co-compact subgroups. Our results only rely on the assumption that either
one of the translation and modulation group (in some cases both) are co-compact
subgroups of the time and frequency domain. This presentation offers a unified
approach to the study of continuous and the discrete Gabor frames.Comment: Paper (v2) shortened. To appear in J. Fourier Anal. App
