Gabor Analysis goes back to a seminal paper by D. Gabor in 1946, where he claims that every function can be expanded in a unique way into a double series of standard Gauss functions, shifted in both time and frequency (i.e. on the Fourier transform side) along the integer lattice. Despite the convincing intuitive heurisitc arguments provided by him the theory of Gabor expansions was not very relevant for a long time, partially due to the lack of mathematical understanding of the problem. It is not just the non-orthogonality of the suggested family which makes the problem interesting and challenging from a mathematical point of view.

After a period of extensive mathematical work in this area (cf. the book on FOUNDATIONS OF TIME-FREQUENCY ANALYSIS by K.Groechenig) we can split the involved problems into smaller subproblems, each of which has a quite understandable and natural answer in the given context.

So we can present a modern view on the subject by making connections to (numerical) linear algebra, the theory of frames and Riesz bases, the appropriate family of Banach spaces of functions and distributions (the so-called family of modulation spaces, introduced 25 years ago by the speaker). For all this not much more than a understanding of a few basic concepts from Banach spaces theory and some ideas from the theory of group representations are relevant.