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The Banach Gelfand Triple and Fourier Standard Spaces
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\centerline{ {\large
Hans G. Feichtinger, Vienna
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Central objects of {\it classical Fourier Analysis} are the Fourier transform (often just
viewed as an integral transform defined on the Lebesgue space $L^1(R^d)$), convolution
operators, periodic and non-periodic functions in $L^p$-spaces and so on.
{\it Distribution theory} widens the scope by allowing larger families of Banach
spaces of functions or generalized functions and extending many of the concepts
to this more general setting. Although, according to A.~Weil the natural setting
for Fourier Analysis (leading to the spirit of Abstract Harmonic Analysis: AHA)
most of the time one works in the setting of the Schwartz space ${\mathcal S}(R^d)$
of rapidly decreasing functions and its dual space, the {\it tempered distributions}.
In this setting weighted $L^2$-spaces and Sobolev spaces correspond to each other
in a very natural way.

In this talk we will summarize the advantages with respect to the level
of technical sophistication and theoretical background which is possible
when one uses instead of the Schwartz-Bruhat space $\mathcal S(G)$ the
Segal algebra $S_0(G)$ and the resulting Banach Gelfand Triple $(S_0,L^2,S_0')$,
which appears to be suitable for the description of most problems in AHA
as well as for many engineering applications (this part is beyond the scope
of the current talk). Among others the use of {\it Wiener amalgam spaces}
$ W(L^p,\ell^q)$ and {\it modulation spaces} $M^{p,q}$
(introduced by the author in the $1980$s) belong to a comprehensive family
of Banach spaces $(B,\| \cdot\|_B)$ embedded between $S_0$ and $S_0'$,
which we call {\it Fourier Standard Spaces}. These spaces have a {\it double
module structure}, with respect to convolution by $L^1$-functions and
pointwise multiplication with functions from the Fourier algebra $FL^1$.
The most interesting examples are Banach spaces of (generalized) functions
containing $S_0(G)$ as a dense subspace and such that time-frequency
shifts $f \mapsto \pi(t,\omega)f$ are isometric on $(B,\| \cdot\|_B)$,
where $$ \pi(t,\omega)f(x) = e^{2 \pi i \omega \cdot x} f(x-t), \quad
x, t, \omega \in R^d,$$
or the dual of such a space. There is along list of examples of such
spaces.