>>11005347>Can someone explain what Fourier transforms are in VERY simple terms?There are two concepts here: Fourier Series and Fourier transforms.

1. Fourier Series: any periodic function F(x+L)=F(x) can be expanded in sum of sines and cosines with the same periodicity (this means the same frequencies or multiples of it). This is because sines and cosines form a complete basis of blah, blah, blah, you don't care about this.

What is the idea? Similar to a Taylor expansion: if you have F(x+L)=F(x), at first order you can say it's approx. A*sin(phi+2*Pi*x/L), with A and phi constants. It's better if we don't use the phi, so let's write it as B*sin(2*Pi*x/L)+C*cos(2*Pi*x/L), where B and C are constants you have to determine. This is the first order of your approximation, akin to f(x)=f(0) in a Taylor expansion.

This is a very poor approximation, so we can improve it adding the next term (like in a Taylor expansion we go from f(x)=f(0) to f(x)=f(0)+f'(0)*x). In this case the next term is D*sin(4*Pi*x/L)+E*cos(4*Pi*x/L), where again D and E are constants you have to determine.

You can continue and, in general, the nth term is given by Bn*sin(2*n*Pi*x/L)+Cn*cos(2*n*Pi/x).

2. Fourier transform: it's a continuous version of the Fourier series. When you take L to infinity, the sum over n becomes a integral and some wonderful properties appear.