It is the purpose of this presentation to recall some of the reasons why engineers and mathematicians are interested in the Fourier transform. As a student I was told, that the Fourier transform is important, because it transforms the complicated operation of convolution into simple pointwise multiplication. But why should we be interested in convolution? A valid answer came to me from the engineers, who are teaching linear time-invariant systems, impulse response and transfer functions in their first courses.

Comparing the two sides of Fourier Analysis, the applied and the theoretical one (which in fact via the theory of tempered distributions by Laurent Schwartz, with important applications to partial differential equations), one observes that they have not too much to do. I will provide a few striking examples in the talk.

Given this situation I suggest a reconciliation of the two worlds, based on a long-term cooperation with engineers, both in research and teaching. The theory of Banach Gelfand Triples, also known as Rigged Hilbert Spaces, provides such a possibility. The modern approach to time-frequency analysis (TFA) allows a simple description of the Segal algebra $S_0(R^d)$, which forms an algebra of test-functions (via integrability of the STFT). The dual space (or equivalently distributional completion), also called space of ``mild distributions'' can be described as the space of all tempered distributions which have a bounded spectrogram (STFT). As time permits we will give a few examples, showing that within this setting a mathematically justified treatment of most expressions arising in engineering applications (such as Shannon's Sampling Theorem, representation of systems as convolution operators, etc.) is possible.

The material presented is part of a long-term project by the speaker, and there is a list of talks and papers available from the NuHAG web-page, e.g.
www.nuhag.eu/talks (access via ``visitor'' and ``nuhagtalks'').