Gabor Analysis is an important branch of time-frequency analysis, going back to the seminal work of D.Gabor (1946), suggesting to discretize the continuous inversion
formula for the short-time Fourier transform (= STFT, also called Siding Window
Fourier transform, or spectrogram). While already used for a long time, e.g. in
the practice of signal analysis or signal processing, its detailed mathematical
analysis and a deeper understanding of the Gabor transform (sampled spectrogram)
and a fine analysis of the properties of operations that can be understood or modelled using Gabor analysis has been carried out only in the last 20 years.

It is the purpose of this presentation to indicate some of the mathematical
progress that has been made, and the explain how it effects the usefulness of Gabor analysis for application areas. Gabor analysis for signals of finite energy (functions
in $L^2(R)$) is complicated by various technical problems, such as the non-orthogonality of its building blocks, which are TF-shifted copies of a given Gaussian pulse, comparable to a micro-tonal piano. We will try to point out how those problems can be separated resp. resolved using basic ideas from linear algebra or group representation theory.

In addition we will point out that even in our daily life (at least that of young people using MP3-players and mobile phones) interesting aspects of Gabor analysis play a role. In particular, we will explain the concept of Gabor multipliers, and how they can be used to model slowly time-variant channels as they occur in mobile communication, between the sending station and the receiver. The fact that multi-path propagation takes place, but that in addition movements of the receiver imply some Doppler effect, allows to describe such linear operators using the spreading representation, which by itself is an interesting mathematical object, and an important tool for the time-frequency analysis of operators.