Distribution Theory by Riemann Integrals

Abstract

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.