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 $x∈[−\pi,\pi]$?

Why don't you do it yourself and see the difference? Do you know how the coefficients of the Fourier series on $[a,b]$ is found? — user66733, Sep 24 '13 at 17:55
He is asking a reasonable, albeit conceptual question, I vote to reopen because I am sure that many would like to see what impact the domain of a function has on its Fourier Series. — laughing_man, Sep 26 '13 at 01:50

score 1 · Answer 1 · answered Sep 24 '13 at 18:13

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.

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

1 Answers1