The Banach Gelfand Triple (S_0,L_2,S_0')(Rd) can be viewed
as a quite universal setting for time-frequency analysis and
in particular Gabor Analysis, but as it has been pointed out
in some recent papers by the speaker it is also useful to
explain questions arising in classical Fourier Analysis.

We call the elements of S_0' ``mild distributions''. One
can characterize them among all tempered distributions as
those which have a bounded STFT (short-time Fourier transform).
As such, the are a suitable reservoir to model e.g. acoustic
signals, which allow to perform (in real time) the computation
of the STFT, so they are natural occuring objects of real life
which ``have a bounded STFT'' (independent of mathematical
considerations).

In order to make this theory accessible for engineers and
physicists and avoiding complicated mathematical concepts
(e.g. Lebesgues spaces, topological vector spaces, concepts
of duality) it is possible to properly describe the concept
of mild distributions, based on the Riemann integral only.
Alternatively, one can show that a sequential approach, in the
spirit of Lighthill is just an equivalent (even more elementary)
description of the same Banach space of distributions. Instead
of the abstract w*-convergence in a dual Banach space one has
to consider simply uniform convergence over compact set in the
STFT domain.