A quick description for engineering students of distributions (generalized functions) and their Fourier transforms

TL;DR

This paper provides undergraduate engineering, physics, and math students with a clear, rigorous introduction to distributions and their Fourier transforms, avoiding advanced functional analysis and Lebesgue integration.

Contribution

It offers an accessible, precise explanation of distributions and Fourier transforms tailored for students with basic calculus and linear algebra backgrounds.

Findings

Clarifies the definition of distributions without advanced prerequisites

Demonstrates how to rigorously perform Fourier transforms on distributions

Provides a practical framework for understanding generalized functions in engineering and physics

Abstract

These brief lecture notes are intended mainly for undergraduate students in engineering or physics or mathematics who have met or will soon be meeting the Dirac delta function and some other objects related to it. These students might have already felt - or might in the near future feel - not entirely comfortable with the usual intuitive explanations about how to "integrate" or "differentiate" or take the "Fourier transform" of these objects. These notes will reveal to these students that there is a precise and rigorous way, and this also means a more useful and reliable way, to define these objects and the operations performed upon them. This can be done without any prior knowledge of functional analysis or of Lebesgue integration. Readers of these notes are assumed to only have studied basic courses in linear algebra, and calculus of functions of one and two variables, and an introductory course about the Fourier transform of functions of one variable. Most of the results and proofs presented here are in the setting of the space of tempered distributions introduced by Laurent Schwartz. But there are also some very brief mentions of other approaches to distributions or generalized functions.