  It is the purpose of this article to outline a course that can be given to
engineers looking for an understandable mathematical description of the
foundations of distribution theory and the necessary functional analytic
methods. Arguably, these are needed for a deeper understanding of basic
questions in signal analysis. Objects such as the Dirac delta and Dirac comb
require a proper definition, and it should be possible to explain how one can
reconstruct a band-limited function from its samples by means of simple series
expansions. It should also be useful for graduate students who want to see how
functional analysis can help to understand fairly practical problems, or
teachers who want to offer a course related to the "Mathematical Foundations of
Signal Processing". The course requires only an understanding of the basic
terms from linear functional analysis, namely Banach spaces and their duals,
bounded linear operators and a simple version of weak$^{*}$-convergence. As a
matter of fact we use a set of function spaces which is quite different from
the collection of Lebesgue spaces used normally. We thus avoid the use of
Lebesgue integration theory. Furthermore we avoid topological vector spaces in
the form of the Schwartz space. Although all tools developed and presented can
be realized on LCA groups, we restrict our attention in the current
presentation to the Euclidean setting, where we have (generalized) functions
over $R^d$. This allows us to make use of simple bounded, uniform partitions of
unity, to apply dilation operators and to make use of special functions such as
the Gaussian. The problems of the overall current situation, with the
separation of theoretical Fourier Analysis as carried out by (pure)
mathematicians and Applied Fourier Analysis (as used in engineering
applications) are getting bigger and therefore courses filling the gap are in
strong need.
