How does $\sum_{j=0}^{\infty} \sum_{l=-\infty}^{\infty} \phi_l(x,t)$ become $\phi_0+\sum_{j=1}^{\infty} \phi_j + \phi_{-j}$?

Asked Dec 27 '18 at 11:51

Active Dec 27 '18 at 11:51

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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?

asked Dec 27 '18 at 11:51

mavavilj

Just separate the terms with $l >0, l<0$ and $l=0$. – Kavi Rama Murthy Dec 27 '18 at 11:54
@KaviRamaMurthy Intuitively yes. However, I would argue that the notation is "slightly" confusing, because e.g. for $j=1$ it would literally read $\phi_0$, $\phi_1$ and $\phi_{-1}$, but there does not exist value for $j=-1$ in the sum. So it's more like an "implicit" meaning of "the negative part of index $j$". – mavavilj Dec 27 '18 at 11:55

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