The functions must be at least $C^2$, preferably $C^\infty$. Completeness for square-integrable functions is sufficient. The conditions at the points $x=0,L$ prescribe the values of the function and its derivatives.
In my case I need simply $f(0)=f(L) = 0$ and $f'(0)=f'(L) = 0$, but I wonder whether one may consider the general case of arbitrarily prescribed values $f^{(n)}(0)$ and $f^{(n)}(L)$?
Does it get easier if I prescribe the values at $x=0$ and demand periodicity, i.e. $f^{(n)}(L)=f^{(n)}(0)$ up to some order $n$? Literature recommendations are welcome. Thank you in advance.