Functions with equivalent Fourier coefficients

Question

Given two functions $f(x),g(x)$ defined over $\mathbb{R}$, satisfying $f(x+P)=f(x)$ and $g(x+P)=g(x)$ for $P\in\mathbb{R}$, consider their Fourier transformation: \begin{eqnarray} \tilde{f}(k)=\frac{1}{P}\int_{0}^{P}dx e^{ikx2\pi/P}f(x), \hspace{10pt} \tilde{g}(k)=\frac{1}{P}\int_{0}^{P}dx e^{ikx2\pi/P}g(x). \end{eqnarray} Now, if $\tilde{f}(k)=\tilde{g}(k)$, what can be said about the functions $f(x),g(x)$? Of course if $f(x)=g(x)$ the relation is clear, but the question is if given two FT of two functions, with equivalent Fourier decomposition, it always holds that $f(x)=g(x)$; I am somewhat inclined to think this is not necessarily the case but I would like to know about this in a direct way, if there are any theorems about it.