Assuming the many of the participants of this conference may
at some point also have to teach a course in Fourier Analysis
I consider it
eminently important that such a course should
satisfy the needs of a modern society, with an increasing
role for digital signal processing, or the simulation of
linear systems using the computer. While Laurent Schwartz'
theory of tempered distributions (almost as old as the
speaker) is well established it is highly demanding in terms
of mathematical background, and mostly relevant for the solution
of PDEs.

This talks should point out that a theory of ``{\it mild distributions}''
can be based on quite simple ideas, and not much more than the
Riemann integral. The key to this modern (and {\it simple}!)
approach to a theory of {\it generalized functions} provides
a solid mathematical basis for objects such as Dirac measures
or Dirac combs. Mild distributions have a Fourier transform,
which is also a mild distributions, and they have a bounded
spectrogram. Such spectrograms can nowadays be produced on
real-time, and in this sense audio-signals are typical examples
of objects which should undergo a ``localized frequency analysis''.

The analogy to the number systems, with the natural embedding
$\QRC$ is used as a motivation and orientation for a sequential
approach. Actual computations are done in the field of rational
numbers $\Qst$, and extended to $\Rst$ by taking limits (e.g.
in order to define $\pi \cdot \sqrt{5}$). Finally a ``trick''
allows to extend the usual calculations to the even larger
(and more flexible) field of complex numbers $\Cst$.

By its genesis the function spaces used, namely the
Segal algebra $\SORdN$ (also called {\it Feichtinger's algebra}),
together with $\LtRdN$, the Hilbert space of square integrable
functions, and the dual space $\SOPRdN$ (of mild distributions)
are very well suited to deal with questions of time-frequency
analysis and discuss the stability of so-called Gabor expansions
of signals. In the audio case this can be viewed as a kind of
graphical score computed from the recorded sound signal.