[Math] Series of squared Fourier coefficients

Question

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?

Answer 1

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 $$ \langle h,h^*\rangle. $$