Fourier standard spaces are Banach spaces (of distributions) on LCA group which are isometrically invariant under time-frequency shifts. We will expore some of the general properties of these spaces and related construction principles. There is a smallest (the Segal algebra SO) and a largest space (the dual of SO) in this class.

Among others it is the purpose of this talk to indicate the richness of this family of spaces, and the role of the two module structures on these spaces, namely the L1 convolution structure and the FL1 pointwise multiplication structure. This goes back to an old paper of the author (with W.Braun).

FULL ABSTRACT
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 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 distibutions in $S_0'(R^d)$, including Dirac measures, Dirac combs, or pure frequencies.

Fourier standard spaces is 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).

It is the purpose of this talk to indicate the richness of this family of Fourier standard spaces, among them Wiener amalgam spaces or 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.