Banach Gelfand Triples for Gabor Analysis

It is the purpose of this survey note to show the relevance of a Gelfand triple which is closely connected with time-frequency analysis and Gabor analysis. The Segal algebra S_0 and its dual can be shown to be - for a large variety of concrete cases - a convenient substitute for the Schwartz space and its dual, the space of tempered distributions. This concrete pair of Banach spaces is actually a Gelfand triple, which allows to describe in a very intuitive way the properties of the classical Fourier transform and other unitary operators arising in the treatment of various mathematical questions, e.g. multipliers in harmonic analysis. We will demonstrate the usefulness of the Banach Gelfand triple within time-frequency analysis, with a special emphasis on questions from time-frequency analysis and Gabor analysis.