The talk is supposed to discuss the following issues:

1) The importance of Gabor expansions is based on the fact that the coefficients of a signal (function or distribution) in a good Gabor system (ideally a tight Gabor frame) contain information about the energy distribution in the given signal over a lattice $\Lambda$ in phase space.

2) This allows to understand the action of operators, among them the most simple ones, the so-called Gabor multipliers. Such operators factorize through a diagonal matrix, whose entries are weights indexed by the lattice points, with the idea of reinforcing or damping them, in a time and frequency variable way, e.g. by filtering out the voice of Caruso from a given old recording.

3) So Gabor systems should make the synthesis and use of Gabor multipliers easy and efficient. There are questions concerning the
computational complexity, the mapping properties between modulation spaces, or the numerical realization of the best approximation of a given operator by a Gabor multiplier have to be discussed.

4) Eigenvalues and eigenvectors for decent Gabor multipliers with smooth symbols should be computable in an approximate way, thus opening up the way to a discussion of (an alternative approach to) discrete Hermite functions.