Apply fourier series to the function $f(x)=\mathrm{sgn}(\cos x)$

Question

There's function:

$$f(x)=\mathrm{sgn}(\cos x).$$

What is it's Fourier series?

I have never worked with this class of functions actually, so I do not know where I should start and how.

Answer 1

Hint.

You have $\mathrm{sgn}(\cos(x))=1$ if $x\in ]-\pi/2,\pi/2[$ (mod $2\pi$) and $\mathrm{sgn}(\cos(x))=1$ if $x\in ]\pi/2,3\pi/2[$ (mod $2\pi$).

The function

$$x\mapsto \mathrm{sgn}(\cos(x))=1$$

looks like this:

You can just apply your formulas for Fourier series, it will work fine. Separate the integrals according to the values $1$ or $-1$ of your function, and the computation will work fine.