Most often Gabor theory is investigated for functions on 
,
i.e., the continuous, non-periodic and one-dimensional case
of Gabor theory is discussed.
Only in the last years alternative settings have been considered.
Gabor expansions for discrete signals can be seen as
part of Gabor theory over 
, whereas numerical
implementations can only work with finite signals. Since
these are naturally
identified with discrete and periodic signals, Gabor
theory over finite cyclic groups
is the appropriate
model for this situation.
The essential ingredients of Gabor theory are the commutative (=abelian)
group of translations in combination with
another commutative group, the so-called dual group of
modulation operators. Hence it is possible to extend Gabor theory to
the general setting of locally compact abelian (LCA) groups 
,
which includes all setting
discussed above. Through the Haar measure
one has a natural 
-space
on , and the existence of sufficiently many
``pure frequencies'', called the
characters of , is assured.
The theory and the formal
computations on all these groups are then the same, and their
derivation becomes somewhat repetitive, often with unnecessary
notation problems.
Independently from the possibility for generalization of the
domain, even for the case
the structure of the group
of all unitary operators (on 
)
generated by the time-frequency shift operators
establishes a connection to representation theory
of locally compact groups
(resp. Lie groups).
Indeed, the study of time-frequency shifts is intimately related
to the study of the so-called Schrödinger representation of
the reduced Heisenberg group. Since this is a three-dimensional
group, with the torus as third component (besides the time and the
frequency parameter), the repeated appearance of exponential terms
of the form 
often has a very natural explanation from
the group theoretical point of view.