The goal of this presentation is to give a short summary
of basic facts concerning time-frequency analysis, in particular
Gabor analysis. It provides the possibility of expanding functions
(and in fact tempered distributions) as unconditionally convergent
double sums of time-frequency shifted atoms, typically some
non-zero Schwartz function, like the Gauss function.

For the analysis of the various mappings (e.g. from the signal
to the sampled short-time Fourier transform) various requirements
have to be made, and a certain Segal algebra $S_0(R^d)$ turned
out to be a versatile tool in this context.


We will indicate that this space is also quite useful for the
context of classical Fourier analysis. Furthermore we point
out how it can be used to establish a so-called Banach
Gelfand triple, consisting of the space $S_0(R^d)$, the
Hilbert space $L^2(R^d) and the dual space, each
one contained in the next one. Among others one
can view the Fourier transform as a unitary Banach
Gelfand triple automorphism of this triple, mapping
pure frequencies into Dirac measures (and being uniquely
determined by this property).


OLD VERSION: Gabor Analysis with general lattices
We plan to describe some progress concerning Gabor
Analysis, using non-separable lattices. This topic
is specially interesting for the case of 2D signals
(applications in image processing), because then
4-dimensional lattices occur.