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

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

In: Mathematics
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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.

A Fourier series is a certain *sum of sinusoids*. Sinusoids are shifted and scaled sine waves. So a particular fourier series might look like:

F(x) = 0.82sin(3x) + 0.13sin(16x+1) + 0.02sin(43x+2)

or it might be an infinite sum of a·sin(bx + c) terms.

Fourier’s famous trick was to show how *any* periodic function, as long as it follows a few rules, can be expressed as a sum like this. This means we can analyze all kinds of data to find periodic components – the ‘frequencies’ in something – which has applications in astronomy, seismology, music, data compression, mechanical engineering, and a zillion other things.

Was it a discovery or an invention? In math, it’s kind of both, you’re kind of discovering stuff *about* what you invent, and inventing stuff based on what you discover. I can’t tell you what gave Fourier this specific idea, but he might have been inspired by Taylor series, which does a similar trick using the terms of a polynomial instead of sinusoids.

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> how can someone come with these ideas?

Fourier’s original motivation came from the study of the [heat equation](https://en.wikipedia.org/wiki/Heat_equation), which models the flow of heat within a material.

This equation is relatively easy to solve in the simple case where the initial heat distribution is a sinusoidal function, but it is not obvious how to solve it for an arbitrary initial condition. However, the heat equation also has the property that it is *linear* – which means that the sum of two solutions of the heat equation is also a solution. Therefore, if you could represent your problem as a sum of simpler problems which you can solve, then you can solve the more complex problem by simply adding them together.

Fourier showed that *any* smooth function could be represented as a (usually infinite) sum of sinusoidal functions, allowing him to solve the heat equation for any initial heat distribution (at least in principle – any actual solution computed this way is only an approximation, because you can’t add up an infinite number of terms, but you can get as close as you need to).