Modelling Signals as Mild Distributions

TL;DR

This paper discusses modeling signals as mild distributions within Fourier analysis, emphasizing practical teaching approaches and applications, including the use of Feichtinger's algebra for representing signals without pointwise definitions.

Contribution

It introduces the concept of mild distributions for signal analysis, providing alternative presentation methods to enhance understanding of Fourier Transform applications.

Findings

Signals modeled as mild distributions like Dirac delta and combs.

Feichtinger's algebra as a framework for measurement signals.

Multiple approaches confirm the robustness of the theory.

Abstract

This note gives a summary of ideas concerning Applied Fourier Analysis, mostly formulated for those who have to give such courses to engineers or mathematicians interested in real life applications. It tries to answer recurrent questions arising regularly after my talks on the subject. It outlines alternative ways of presenting the core material of Fourier Analysis in a way which is supposed to help students to grasp the relevance of this transform in the context of applications. Essentially we consider functions in $S_0(R^d)$ (known as Feichtinger's algebra) as possible measurements, and the elements of the dual space (which can be also described by various completion procedures) is thus the collection of all "things" (in the spirit of signals) which can be measured in a reasonable way. We call them mild distributions. In other words, we want to base signal analysis on the mathematical theory of mild distributions. They do not need to be defined in a pointwise sense. Dirac's Delta or Dirac combs are natural examples. The are "as real as point masses in physics". Various equivalent approaches (including a sequential approach) help to verify the relevant results. The material is based on the experiences gained by the author in the last 30 years by teaching the subject in an abstract or application oriented manner, at different universities, and to audiences with quite different background. For related course material see www.nuhag.eu/ETH20.