The fact that Gabor families (both Gabor frames and Gabor Riesz Basic Sequences) was long considered an obstabcle for its use in signal processing. Convergences and uniqueness issues arose and made the subject look like deep in the domain of functional analysis in order to be understood properly.

By comparing the concepts in Gabor analysis (in particular the concept of Banach frames, for families of Banach spaces) with related, basic concepts in linear algebra (generating systems, pseudo-inverse, linear independent families) resp. numerical analysis one can build up a theory which is quite flexible and easily to handle.

Similar concepts arose already in the theory of coherent states, Anti-Wick calculus, Berezin transforms, Fock spaces, but we will describe them from scratch, with a special emphasis on the role of time-frequency analysis (with symmetric roles for the time and the frequency variable), signal processing and pseudo-differential operators (Kohn-Nirenberg symbols, spreading function, etc.) and Gabor Theory in its many facets, from the fine description of distributions or smoothness of functions, to the numerical implementation using MATLAB.