Can you define a Fourier series for $f(x)$ for any interval you want, even for which $f(x)$ is not periodic on that interval?

Question

Say you have a function $f(x)$, and you generate a Fourier series for it on the interval $-L \leq x \leq L$, and $f(x)$ is piecewise continuous on that interval.

Say also that $f(x)$ is a periodic function but it is not periodic with the interval $2L$, does the Fourier series then still converge to $f(x)$ for all $x$ in that interval where $f(x)$ is continuous?