Hi, if the Fourier series development of $g(t)$ (periodic, $C^\infty$) is 


$$
g(t)=\sum\_{-\infty}^{+\infty}a\_n e^{in\omega t}
$$


does the series


$$
\sum\_{-\infty}^{+\infty}\frac{a\_n^2}{n^2}?
$$
converges toward something known like average $g^2$ or something like that?



Series of squared Fourier coefficients
Assume $a\_0=0$, which easily can be arranged by adding a constant to $g$.
Then the function
$$
h(t)=\frac1{i\omega } \sum\_{n}\frac{a\_n}ne^{in\omega t}
$$
is the primitive of $g$.
Let $h^\* (x)=\overline{h(-x)}$, then the sum you asked for equals the inner product 
$$
\langle h,h^\*\rangle.
$$


