Question: Let
$f(x) = \left\{ \begin{array}{ll} x & \mbox{if } -1 \leq x < 0 \\ x^2 -1 & \mbox{if } 0 \leq x \leq 1 \end{array} \right.$
Define the Fourier serie of f in $[-1,1]$.
Question: I'm learning about that topic and I had have a lot of doubts about the correctness of some mathematical manipulation and theorem's application. For instance, in that problem: may I define the fourier serie of $G(x)=x$ and $H(x)=x^2+1$ and extend those to fourier serie of f, once $f(x)$ is piecewise smooth function? Otherwise, the calculation of the serie will take me a quite long time.
Likely my doubt is about a very wrong possibility, but i'm having a really bad time with it. I don't want the Fourier serie, only the way to solve it.
Your suggestion with $G$ and $H$ won't work. You really just have to bite the bullet and grind out the integrals. By the time you're done, you should be an expert at integration by parts. (Way more productive than watching TV or posting on Facebook.)