With respect to the so-called Short-Time Fourier Transform the family of modulation spaces plays a role comparable to that of the family of Besov-Triebel-Lizorkin spaces with respect to the continuous wavelet transform. In both cases membership of a given function or tempered distribution on one of those spaces can be characterized by the fact that the continuous transform is in some solid Banach space of measurable functions on the domain. Typically weighted mixed-norm spaces are used. By viewing the domain the continuous transform as a group (the "ax+b" group in the case of the wavelet transform, or the so-called phase space in the case of the STFT), one also has to assume that the function space on the group is translation invariant, which makes the characterization of the spaces independent from the particular ``admissible'' mother wavelet or atom used.

The so-called coorbit theory developed by Feichtinger and Groechenig in the late eighties provides a unified view point to these spaces. Specifically modulation spaces can be characterized as (generalized) Wiener amalgam space conditions on the Fourier transform side.

Taking the most simple of those space, namely the Segal algebra $S_0(G)$, characterized by an $L^1$-condition in phase space, $L^2$ and the
dual space $S_0'(G)$ one has a so-called Banach Gelfand triple. It turns out that the concept of Banach Gelfand triples is very flexible, and very well suited in order to describe the subtelties of the Fourier transform (where e.g. summability methods are required to prove the Fourier inversion theorems, or some extra assumptions have to be made in order to ensure the validity of Poisson's formula). Among others, the (generalized) Fourier transform is the only Banach Gelfand Triple isomorphism which maps the ``pure frequencies'' (or characters) into the corresponding Dirac measures.