Although it appears to be quite natural to define the Fourier transform as an integral transform on the space of Lebesgue integrable functions L^1(R^d) even the classical treatment indicates some problems, e.g. for the realization of the inverse Fourier transform, or in it's L^2-version, known as Plancherel's theorem. Nevertheless it is precisely this setting which allows to express that the Fourier transform is like a change from one orthonormal basis to another one, i.e. a unitary mapping.

We will explain in our presentation how a new point of view can be established, making use of a so-called Banach Gelfand Triple, which arose in the field of time-frequency analysis. It serves quite well in order to formulate problems in classical Fourier analysis as well as in Gabor analysis or more generally time-frequency 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.