Motivated by recent formulations of Gabor theory for periodic and for discrete signals, this chapter develops several aspects of Gabor theory on locally compact groups.
First an uncertainty principle in terms of the short time Fourier transform is derived (Lieb's inequalities). It captures the intuition that any signal occupies a region in the time-frequency plane of area at least one. Secondly, the Zak transform, introduced on locally compact abelian groups already by A. Weil, is used to analyze Gabor frames in the case of integer-oversampled lattices in the time-frequency plane. In this context it is observed that the Balian-Low theorem depends on the group structure and that the known versions do not hold for discrete and compact groups. In the final section a notion for the density of lattices in defined and necessary conditions for lattices in the time-frequency plane to generate Gabor frames are derived.