When one looks at the Fourier transform in the mathematical literature the description starts usually with Fourier Series for periodic functions or right away with the Fourier transform as an integral transform. In either case the transform requires to use integrals, and of course the Lebesgue integral appears to provide the natural domain, namely the space L^1 of integrable functions. Similar arguments apply to the convolution integral. Combining the two concepts one can then derive the all-important convolution theorem, Fourier inversion and Plancherel's theorem,
showing that the ``complicated convolution'' is turned into easy pointwise multiplication. But why should we be interested in convolution? Is it a natural product for integrable functions? And which functions do ``have a Fourier transform''?

Aside from heuristic manipulations, leading to the forward and inverse Fourier transform the above results are certainly important to (electrical) engineers, when they deal with translation invariant systems, which are usually described by black boxes. They correspond to convolution operators with the so-called impulse response, which is the output of the system to a ``Dirac delta-function'', and can be described alternatively by their transfer function.

We will describe a mathematically correct approach to convolution and Fourier transform which is based on simple functional analytic principles and encompasses the two aspects of basic Fourier analysis in a way (hopefully well) understandable for both sides (mathematicians and engineers). Furthermore, many aspects of this approach can by supported by a set of simple MATLAB experiments, which connects the material with basic concepts from linear algebra and polynomials with complex coefficients.