**ABSTRACT**
### Aspects of Gabor analysis on locally compact abelian groups

#### K. Gröchenig

##### Department of Mathematics, University of Connecticut, Storrs, Connecticut

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.