How does $\sum_{j=0}^{\infty} \sum_{l=-\infty}^{\infty} \phi_l(x,t)$ become $\phi_0+\sum_{j=1}^{\infty} \phi_j + \phi_{-j}$?
Where $\sum_{l=-\infty}^{\infty} \phi_l(x,t)$ is Fourier series (or in that sense) of $\phi_j$ without the exponential (for simplicity).
"Naively" I read that the latter is a "shortcut" for saying that:
For each $j$ take $\sum_{l=1}^{\infty} \phi_l$ and $\sum_{l=-1}^{-\infty} \phi_l$ and $\phi_0$ to produce $\phi_j$.
However, is my intuition right?