Fourier Series: An Introduction
Chris White
Fourier analysis is one of the most powerful tools in modern mathematics. The (seemingly simple) idea of a Fourier series has surprisingly deep connections to almost every area of math, from representation theory to signal analysis, and can even be used to provide an elegant explanation of the Heisenberg Uncertainty Principle. In this talk, I hope to provide a basic introduction to Fourier series. We will begin by reviewing some concepts, such as Hilbert spaces, which are needed to understand what Fourier series are all about. I will then state the Plancherel Theorem and use it to provide a simple solution to the classic "Basel problem," a problem which went unsolved for almost a century (we will solve it in about 5 minutes).
