This is not a technical, but a {\it conceptual talk}, with the
goal to describe a mindset that should help to better understand
important aspects of modern Fourier and Time-Frequency Analysis.
New problems (such as Gabor expansions) require new tools, in
our case new function spaces, such as Wiener amalgams or
modulation spaces.

It turned out that these spaces, specifically the
Fourier invariant Segal algebra
$S_0(R^d)$ and its dual $S^*_0(R^d)$ (endowed with the $w^*$ topology)
also provide a much more convenient
tool for the description of the Fourier transform than classical
$L_p$-theory. At can be developed without reference to the theory of tempered distributions developed by Laurent
Schwartz, but is in fact a (simplified)
version of it, sufficient for most engineering applications.
Therefore we call members of $S^*_0(R^d)$ {\it mild distributions}.


It can be used to derive key results, like the Shannon Sampling Theorem
or the description of linear, time-invariant systems as convolution operators by some impulse response (or via a transfer function on the Fourier transform
side). The setting of mild distributions allows correctly to handle
Dirac combs, discrete periodic measures, or $L^p$-spaces on an equal footing
and shows how to (correctly) turn typical heuristic statements found in books
on Fourier analysis into correct claims, using distributional convergence (resp.\ $w^*$-convergence) inside of $S^*_0(R^d)$.

Thus the setting of the Banach Gelfand Triple $(S_0,L_2,S^*_0)(R^d)$
appears to be the appropriate setting for all this, because it
allows to justify mathematically what engineers and physicists
are doing, but also because it can be used to support computational
schemes. Finally the setting grew out of long-standing attempts
to develop a theory of extended Fourier transforms over LCA groups.

The talk comes with a number of illustrations, showing via
suitable pictograms for the different function spaces what the
correct inclusion relations are between them. At the technical
level we only need basic functional analysis and a bit of group theory.
Please take it as a showcase for an extensive new landscape, allowing
you to take just a glimpse into a novel approach to what I call
{\it Conceptual Harmonic Analysis} (more then Abstract $+$ Numerical
Harmonic Analysis).