I came across this question when trying to prove that the composition of Fourier series and its inverse operation will give back the original function. Let $A(\theta)$ be a period-$L$ function. Define the frequency $\ell = 2\pi k/L$, then, \begin{align} A(\theta) &= \sum_{k=-\infty}^\infty A_\ell e^{i \ell \theta} \\ &= \sum_{k=-\infty}^\infty\frac{1}{L} \int_0^L d\theta' A(\theta') e^{-i \ell \theta'} e^{i \ell \theta}\\ &= \frac{1}{L} \int_0^L d\theta' A(\theta') \sum_{k=-\infty}^\infty e^{i (2\pi k /L) (\theta-\theta')}\\ &= \int_0^L d\theta' A(\theta') \sum_{k=-\infty}^\infty \frac{1}{L} e^{i 2 \pi k (\theta-\theta')/L} \\ &= \int_0^L d\theta' A(\theta') \sum_{m=-\infty}^\infty \delta\left(\theta-\theta'+mL\right) \end{align}
where in the last step I used the Poisson summation formula. However I cannot conclude the result is the original function at the boundary point $\theta = 0, L$ (since I will have 2 delta functions in the summation that affects the integral). I am wondering if there is anything wrong in this argument, and does how we define $A(\theta)$ at the boundaries have anything to do with this issue (e.g., if the function is 0 at the boundary then everything seems fine)? Thank you in advance!