At the beginning of many courses on Fourier Analysis one often finds
a discourse of the Lebesgue space $L^1(R^d)$ as a starting point,
because both for the definition of the Fourier transform as well as
the definition of the convolution of two functions integrability
appears to be indispensable. On this basis the all important
{Convolution Theorem} can be shown:
$$ \widehat {f \star g} = \widehat f \cdot \widehat g, \quad \quad f,g \in L^1(R^d). $$

We are proposing an alternative approach, inspired by the theory of Banach modules and on the other hand by the view of engineers on translation invariant linear systems. The general setting will make use the so-called Banach Gelfand Triple  $(S0,L2,SO*)$, with $S0$ as a Banach algebra of test functions (which is Fourier invariant), and the dual space, meanwhile called the space of mild distributions (with both the norm and the w*-topology).

The talk is essentially summarizing the experiences of the speaker bz contacts with engineers and physicists.