This talk will describe the basic principles of Gabor analysis, a very natural form of local time-frequency analysis (which in a way is even in the background of the MP3 compression algorithm). The starting point is a discretized (along with some suitable lattice) version of the so-called short-time (or sliding window) Fourier transform (STFT). Usually, the window or localizer is taken some Gauss function, but it
can be any Schwartz function or even more general.

We will explain, why the treatment in the pure Hilbert spaces setting is not satisfactory, we will point out that there is a so-called Banach Gelfand triple named $S_0,L_2,S_0')$, consisting of a Banach space of test functions, densely embedded into the Hilbert spaces $L2(Rd)$ and in turn continuously embedded into the dual space of $S_0$, which consists of so-called ``mild distributions''.

In addition to various characterizations of these function spaces, we will demonstrate their usefulness in the context of classical Fourier analysis as well as for the description of Gabor multipliers. These operators
are connected with the atomic decompositions of functions or distributions arising from the collection of time-frequency shifted windows appearing in the description of the sampled STFT. In the context of engineering applications, they arise as time-variant channels and play a role in mobile communication.