Let $\phi$ be an eigenfunction of the Laplacian $\Delta$ on the $n$-torus $T^n$, with eigenvalue $-\lambda$, i.e. $\Delta \phi + \lambda \phi =0$, then :
$$ \phi (x)= \sum_{|n^2|=\lambda} \hat{\phi} (n) e^{in \cdot x}.$$
Why can we write $\phi$ this way?
