Fourier analysis was initially introduced as a way to study the thermodynamic heat equation. At its simplest, it is the study of how to represent functions as an infinite sum of sines and cosines. However, the evil mathematicians felt that Fourier analysis was too simple and decided to steal it from the hardworking physicists and expand on it. In addition, they decided to complicate it by introducing Harmonic analysis. So, what is the difference between Harmonic and Fourier analysis? We say that Harmonic analysis is the process of representing functions on a locally compact group G as a sum of the characters of the group, and Fourier analysis restricts this process to abelian groups.

In this talk I will introduce the Fourier series, the Fourier transform and how it applies to abelian groups. What about non-abelian groups, you might ask? The answer is Harmonic analysis! Finally, I will explain the role of Fourier transform in quantum mechanics, specifically how the Fourier transform interchanges position and momentum space.