Let be $f: \mathbb{R} \longrightarrow \mathbb{R}$ an absolutely integrable function, periodic with fundamental period $2L$ and $T$ an arbitrary period (not necessarily the fundamental period) of $f$. Show that
$$a_n = \frac{2}{T} \int^T_0 f(x) \cos \left( \frac{2n \pi x}{T} \right) dx, \ b_n = \frac{2}{T} \int^T_0 f(x) \sin \left( \frac{2n \pi x}{T} \right) dx.$$
I would like to receive a hint, because I don't have idea how to start in order to proof this. Thanks in advance!
$\textbf{EDIT}$
The definition of Fourier Coefficients for a function periodic with period $2L$:
$$a_n = \frac{1}{L} \int^L_{-L} f(x) \cos \left( \frac{n \pi x}{L} \right) dx, \ b_n = \frac{1}{L} \int^L_{-L} f(x) \sin \left( \frac{n \pi x}{L} \right) dx.$$