The usual description of the Fourier transform (over)emphasizes the role of L^p-spaces. Ideas from the theory of Banach modules (M. Rieffel), time-frequency analysis (in particular Gabor Analysis) and Coorbit theory (in particular the theory of modulation spaces) allows to describe a large variety of function spaces which are suitable for the description of operators arising in this context, but also pseudo-differential operators.

The L^p-spaces are a good starting point (to define the Fourier transform as an integral transform, or properly define convolution integrals; or in order to prove Plancherel’s theorem using L^2, which is halfway between L^1 and its dual space L^infty), but they behave not so well under the Fourier transform. On the other hand Sobolev spaces or Schwartz’s tempered distributions are needed in the theory of PDE, but they can be described also by modulation spaces.

The newly introduced family of Fourier Standard Spaces is the (Fourier invariant) family of all Banach spaces of (tempered) distributions which have a double module structure, namely over L^1 with respect to convolution and over the Fourier algebra FL^1 under pointwise multiplication. They are all embedded into SO’ and contain the Segal algebra SO. These spaces can be described over general LCA groups. The talk will restrict to attention to the Euclidean case: G = R^d.