Hilbert spaces appear as a natural extension of the Euclidean space, equipped with a scalar product. They are extremely useful and universal tools in many branches of analysis, and also used heavily in application areas. Among others one has the Riesz representation theorem, allowing to identify in a natural way the set of linear functional with the elements of the original Hilbert space. Unitary isomorphism can be described as the transition from one orthogonal basis to another, and so on.

Unfortunately already the classical Fourier transform in its natural is a unitary automorphism of $L^2(R^d)$, but the natural
interpretation - similar to the case of the FFT - as a change of basis from unit vectors to pure frequencies (none of them belong to $L^2(R^d)$!) is not possible. And despite the fact that the Fourier transform is originally defined as an integral
transform one has to apply summability methods in order to verify the Fourier inversion or Plancherel's theorem. A lot of hard analysis and functional analysis has been developed during the last century to handle to upcoming problems.

It turns out the TF-analysis not only poses a number of delicate questions, but also opens the way to a new Banach Gelfand Triple $(S_0(G),L^2(G),S_0^\prime(G))$ (i.e. a triple, consisting of a Banach spaces of test-functions $S_0(G)$, discovered by the author in 1979, and its dual space, with the Hilbert space $L^2(G)$ in the middle).

It is the purpose of this talk to demonstrate how easy it is to use this Banach Gelfand Triple, how it helps to give a precise mathematical meaning to terms arising in Gabor analysis, but also how various classical questions around the
Fourier transform, multiplier problems, Poisson's formula or sampling problems.