{eli5} What is the Fourier series used for and how does it work?

Question

Also how such inventions are created or discovered? how can someone come with these ideas?

In: Mathematics

Answer 1

First of all, there is no “The Fourier series.” Rather, a Fourier series is any function that fulfils a specific set of requirements. In this case, a Fourier series is any periodic function that is represented by a weighted sum of sine and cosine functions, and up to an infinite series of them. The main takeaway from this is that this function can be constructed from solely sine and cosine.

This sounds quite unassuming at first glance, but let’s take a little glance at what this means. Sine and cosine are relatively special functions. They oscillate from between -1 and 1, and have a continuous derivative and continuously repeat. This means that they are inherently perfect for storing periodic data. Thus, the Fourier series, and indeed, any Fourier method has a widespread application in anything that involves sending periodic data. Imagine you’re trying to send a continuous, periodic sound through a phone. You would have to continuously send a snapshot of the sound wave at every moment, essentially eating up infinite bandwidth over time. But with the magic of a Fourier series, you can simply deconstruct the signal, and send over the data as a series of coefficients of functions of sine and cosine! This has the effect of compressing your data massively, and a sometimes intended side effect of removing noise, as you are essentially approximating it by its closest representation possible with some number of combinations of sine and cosines added together. Modern telephones use series summed into the millions.

The Fourier series isn’t what one would conventionally consider as an “invention,” but rather a derivation of representation consistent with the fundamentals of mathematics. Josef Fourier needed a way to describe how heat transfers into a metal plate in his 1807 publication *Mémoire sur la propagation de la chaleur dans les corps solides* (Treatise on the propagation of heat in solid bodies). Through trying to do this, he established the fact that any function can be represented by a trigonometric series. Ideas like this are not things that one simply “comes up” with, rather they are things born through years of hard work, usually an ambition towards some sort of an end, and a fair bit of luck.