Fourier series of $f(x)=x^2$ in $x∈[0,2\pi]$ and in $x∈[−\pi,\pi]$?

Question

Fourier series - what is the difference between the Fourier series of $f(x)=x^2$ in $x∈[0,2\pi]$ and in $x∈[−\pi,\pi]$?

Answer 1

Conceptually, Fourier series try to express a periodic function in terms of sines and cosines. Consider the two situations you mentioned here, letting $f(x) = x^2$.

If your periodic function is $f$ from $[0,2\pi]$, then it would be just the right part of the parabola from 0 to $2\pi$ repeated again and again.

But if you evaluate f from $-\pi$ to $\pi$, then the function looks like a full parabola from $-\pi$ to $\pi$, which, if repeated, creates a completely different periodic function than the other case.

Therefore, the domain is very important in defining what exactly the periodic function you are trying to express in Fourier's domain.