Group theory as unifying language

Most often Gabor theory is investigated for functions on $ \R $, 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 $ \Z$, 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 $ \cal G$, which includes all setting discussed above. Through the Haar measure one has a natural $ \Ltsp$-space on $ \cal G$, and the existence of sufficiently many ``pure frequencies'', called the characters of $ \cal G$, 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 $ \cal G = \Rst$ the structure of the group of all unitary operators (on $ \LtR$) 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 $ e^{iax}$ often has a very natural explanation from the group theoretical point of view.