The talk will indicate that there are many different motivations to introduce convolution, but also the Fourier transform, as a unitary transformation which diagonalizes convolutions (or: the Convolution Theorem demonstrates that the FT turns convolution into ordinary pointwise multiplication).

The theory of Rigged Hilbert Spaces, resp. the specific Banach Gelfand Triple, based on the Segal algebra S_0 comes in handy in order to describe the properties of the Fourier transform and to make the analogy between the discrete setting (this is based on the DFT, the discrete FT, which is implemented as the FFT, the Fast Fourier Transform). As it will be explained, there are several new and simplified approaches to this Banach space of test functions and its dual, which is meanwhile known as the space of ``mild distributions''. It can be characterized as a kind of completion of the space of test functions, but also as a subspace of tempered distributions with bounded spectrogram.