Here is the statement :
Let $f:\mathbb{R}\to \mathbb{C}$ a continuous map which is $\mathcal{C}^1$ by pieces and such that $f\in \mathcal{L}^1(\mathbb{R})$. Moreover, $\hat f \equiv 0$ in $\mathbb{R}-[-1/2,1/2]$. Then for all $x\in\mathbb{R}-\mathbb{Z}$, $$f(x)=\frac{1}{\pi}\sum \limits_{n=-\infty}^{+\infty}\frac{\sin(\pi(x-n))}{x-n}f(n).$$
Thanks in advance !