{{ The role of Banach Gelfand Triples for Conceptual Harmonic Analysis}}

{ Hans. G. Feichtinger (TUM/NuHAG) }

Classical Harmonic Analysis is focussing very much on the Lebesgue spaces $L^1,L^2,L^\infty$, because they appear at first sight as natural domains for convolution or the Fourier transform. As it has turned out a variant of
distribution theory, arising from problems in time-frequency analysis, gives rise the a description of the Fourier transform as an automorphism of the {\it Banach Gelfand Triple } (or rigged Hilbert space) $(S_0,L^2,S_0')(R^d)$, i.e. the Plancherel theorem restricts well to the space of test functions $S_0(R^d)$ but also extends well to the distributions in $S_0'(R^d)$, including Dirac measures, Dirac combs, or pure frequencies.

As time permits we will also talk about
{\it Fourier Standard Spaces}, a family of Banach spaces between $S_0(R^d)$ and $S_0'(R^d)$, with some extra properties, essentially allowing smoothing (by convolution) and localization (by pointwise multiplication), indicating the richness of this family of Fourier standard spaces, among them {\it {Wiener amalgam spaces}} or {\it modulation spaces}, and to present a few general claims which can be made for the Banach spaces in this family. Of course, the classical $L^p$-spaces belong to this family, however without playing a significant role there.