Gabor Analysis can be performed for functions or distributions over general LCA (locally compact Abelian) groups. It goes back to D.Gabor's seminal paper of 1946, where he proposed to expand ``arbitary L2-functions'' uniquely as double series of Gaussian atoms, moved along the integer lattice Z x Z by so-called time-frequency shift. For audio-signals one could interpret this as a statement, that one can produce arbitrary sounds (of finite energy) from elementary "tunes", with a constant spacing in time and frequency.

Modern Gabor analysis is describing how function space properties can be translated into properties of coefficients (decay and summability conditions) and vice versa, using Gabor systems in $R^d$, obtained by means of general lattices L in R^{2d}. Frame theory, Riesz basic sequences, and Gabor multipliers are objects that are studied in this context, modulation spaces are relevant for the treatment of certain pseudo-differential operators, while underspread operators model problems in mobile communication.

As time permits we will also explain what is known about Gabor analysis in 2D, what kind of numerical methods can be based on the algebraic and functional analytic structure regular Gabor systems, and how one can perform space-variant filtering of images using 2D Gabor systems.