\forall f\in C^\infty\left(\left[-\frac{T}{2};\frac{T}{2}\right]\right),
  \forall t\in \left[-\frac{T}{2};\frac{T}{2}\right],
  f(\tau) = \sum_{k = -\infty}^{+\infty} e^{2i\pi\frac{k}{T}t} \times
  \underbrace{\frac{1}{T}
              \int_{-\frac{T}{2}}^{\frac{T}{2}} f(t) e^{-2i\pi\frac{k}{T}t} dt
              }_{a_k = \tilde{f}\left(\nu = \frac{k}{T}\right)}