I'm having trouble visualizing the Fourier series of $x \sin(x)$, say if it's graphed over $[0,4 \pi]$ and then repeated.
This is an integrable function, so I should be able take the Fourier series right? I'm just having trouble with the intuition here.
I know the Fourier transform gives a picture of 'how much' of each frequency of sin functions is used in the construction of the function as a Fourier series. Each 'loop' could be thought of as a sum of sines with different frequencies, and so this would certainly not be a Dirac impulse like $sin(x)$.
I was initially worried about the lack of differentiability at each integer multiple of $4 \pi$ but realized that's nonsense, as a saw tooth function has a Fourier series.
Can someone clear up my confusion? Sorry if I'm casually throwing around incorrect terminology; I haven't had to think about Fourier series for over a year.