Consider the function $$H:[-3,2] \to[2,2] $$ \begin{cases} -2 & -3\leq x < \frac{-1}{2} \\ 2x & \frac{-1}{2}\leq x< \frac{1}{2} \\ 2 & \frac{1}{2}\leq x \leq 2 \end{cases} Determine the Fourier representation of H.
My question is: since this is not over a symmetric interval, how would I go about representing this function as a Fourier series? Any help would be appreciated!
For a Fourier series you do not need symmetric intervals. You can just scale an translate the trigonometric polynomials in such a way that they become periodic wrt. to any given (bounded) interval. If you do that you have to ensure that you also get an orthonormal system by introducing appropriate factors in front of either your Fourier integral or the series or both. (Like $1/pi$ in the classical case).
In your particular example just start out with $\sin$ and translate and scale it so that it vanishes at $-3, -1/2$ and $2$.
Alternatively, translate and scale the function which you need to develop into a series to $(-\pi, \pi)$, calculate it's Fourier series and translate back.