It will be explained, that a Banach Gelfand Triple, which arose in the field of time-frequency analysis, servers also well in order to formulate problems in classical Fourier analysis. The so-called Segal algebra S_0(Rd) (it can be defined even for general LCA groups) shares many properties of the Schwartz space of rapidely descreasing functions, in particular its Fourier invariance. Hence the ordinary Fourier transform, defined via Riemannian integrals, extends naturally to the dual space, S_0'(Rd), which should be seen as a Banach space, but also as a space carrying the w*-topology (also easily interpreted). In between those two spaces on finds the Hilbert space L2(Rd). Together they form a Banach Gelfand Triple. Since the Fourier transform leaves each of them invariant it is a perfect example of a Banach Gelfand triple isomorphism (or even automorphism). Other examples (e.g. the spreading function representation) will be given. Overall, questions of summability, the representation of shift-invariant operators on function spaces as convolution operators resp. as Fourier multipliers also becomes an easy task under this view-point.