This talk is meant to promote the setting of so-called Fourier Standard Spaces, a family of Banach spaces of (mild) distributions which have a double module structure, namely with respect to convolution (smoothing) and pointwise multiplication (localization).

Although Fourier Standard Spaces are naturally defined over LCA (locally compact Abelien) groups we choose the describe the results here in the Euclidian context $G = R^d$. Here Fourier Standard spaces (FouSSs) can be described as Banach spaces of tempered distributions which have a double module structure, namely with respect to convolution over the Banach algebra $L^1(R^d)$, and with respect to pointwise multiplication over the Fourier Algebra $FL^1(R^d)$.

Any such Fourier standard spaces allows a continuous embedding of the Segal algebra $S_0(R^d)$ (also known as Feichtinger's algebra meanwhile) and is contained in the dual space $S'_0(R^d)$, by definition called the space of ``mild distributions'' over $R^d$. Due to the Fourier invariance of this pair of Banach spaces, combined with the double module structure of FouSSs a number of operations relevant for Fourier analysis and more generally Time-Frequency Analysis can be performed in a natural way.

We will discuss a few of them, like the characterization of multipliers between such spaces (using the engineering terminology of ``impulse response'' and ``transfer function''), the double-module diagram described by the author in 1983, or the derivation of the bounded approximation property for minimal FouSSs (the ones, which contain $S_0(R^d)$ as a dense subspace.