Normally the Fourier transform is defined for L1-functions or tempered distributions. For LCA (locally compact Abelian) groups one has to resort to the even more complicated Schwartz-Bruhat space. The talk will indicate how the Banach Gelfand triple viewpoint, using a Banach space of test functions, namely the Segal algebra S0(G), the Hilbert space L2(G) and the dual space SO'(G) can be used to properly define the Fourier transform in a generality suitable for most engineering applications, but also for the purpose of abstract harmonic analysis. The talk is meant to provide a first overview on the definition and basic properties of this Banach Gelfand triple and how it can be used to prove typical results (e.g. the Shannon sampling theorem for band-limited signals).