It is the purpose of this presentation to explain certain aspects
of Classical Fourier Analysis from the point of view of
{\it distribution theory}. The setting of the so-called {\it
Banach Gelfand Triple} $(S_0,L^2,S_0')(R^d)$ starts from a particular Segal algebra $S_0(R^d)$ of continuous and Riemann integrable functions. It is Fourier invariant and thus an extended Fourier transform can be defined for $S_0'(R^d)$, the space of
so-called {\it mild distributions}. Any of the $L^p$-spaces
with $1 \leq p \leq \infty$ contains $S_0(R^d)$ and is embedded
into $S_0'(R^d)$.

We will show how this setting of {\it Banach Gelfand triples} resp. {\it rigged Hilbert spaces} allows to provide a conceptual appealing approach to most classical parts of Fourier analysis. In contrast to the Schwartz theory of tempered distributions, it is expected that the mathematical tools can be also explained in more detail to engineers and physicists.