Joseph Fourier demonstrated that an arbitrary T-periodic function $f(t)$ can be written as a linear combination of harmonic complex sinusoids:
$$f(t) = \sum_{n=\infty}^\infty c_ne^{iw_ont}$$
(And likewise as $t -> \infty$ or in other words for the case of Fourier Transform, with some limitations).
Where is the proof/demonstration that shows that the series/transform indeed converges to f(t) and thus the identity exists. Further can it be seen that the solution for coefficients $c_n$ are unique from the derivation of the $c_n$. (Derivation: http://cnx.org/contents/8YnJdzjg@8/Derivation-of-Fourier-Coeffici)
It's not so simple: you need some restrictions on the function. If $f$ is continuous and has bounded variation, the series converges pointwise everywhere to $f(t)$. But there are continuous functions whose Fourier series diverge at infinitely many points.