While mathematicians like to work with abstract, continuous
models engineers argue that there is nothing like this on the
computer, and therefore one has to work with discrete, in fact
nite length, i.e. periodic and discrete signals.
It is the purpose of this talk to shed some light on the highly
non-trivial connection between these two worlds.


original plan was this (has been changed):

Gabor Analysis can be realized naturally over any LCA (locally compact Abelian) group. Naturally it is described in the context of Euclidean spaces (with natural notions of time- and frequence shifts) using the continuous Fourier transform, or in the setting of finite groups, resp. in the context of discrete and periodic signals using the corresponding FFT in order to realize the corresponding signal expansions.

In this talk we will report about the challenge to approximate an given continuous problem in different ways, either by trying to perform the necessary computations with the necessary precision in a continuous context, or by finding a ``similar'' discrete context where the standard algorithms (by now availalbe) for regular Gabor families can be applied.

In this connection Cauchy-conditions on such families, i.e.
the question when two approximations to the same continuous situation, with signals of different finite length, can be considered to be (more or less closely) related, not only in a practical sense (based on visual inspection), but also in terms of strict mathematiccal terminology allowing to compute quantitative measures of similarity.