3. Distribution Theory by Riemann Integrals

It is the purpose of this article to outline a syllabus for 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 the Dirac comb should have 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 mathematics 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” at their institutions. 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 \( w^{*} \)-convergence. As a matter of fact, we use a set of function spaces which is quite different from the collection of Lebesgue spaces \((L^p(\mathbb {R}_d),\Vert .\Vert _p)\) used normally. We thus avoid the use of the Lebesgue integration theory. Furthermore, we avoid topological vector spaces in the form of the Schwartz space. Although practically all the tools developed and presented can be realized in the context of LCA (locally compact abelian) groups, i.e., in the most general setting where a (commutative) Fourier transform makes sense, we restrict our attention in the current presentation to the Euclidean setting, where we have (generalized) functions over \(\mathbb {R}^d\). This allows us to make use of simple BUPUs (bounded, uniform partitions of unity), to apply dilation operators and occasionally to make use of concrete special functions such as the (Fourier invariant) standard Gaussian, given by \(g_0(t) = \exp (- \pi \vert t \vert ^{2})\). 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 bigger and therefore courses filling the gap are in strong need. This note provides an outline and may serve as a guideline. The first author has given similar courses over the past years at different schools (ETH Zürich, DTU Lyngby, TU Munich, and currently Charles University Prague) and so one can claim that the outline is not just another theoretical contribution to the field.