I am working on analyzing special electrical circuits in frequency domain. So I need to be able to have the product of several functions represented as Fourier series.
Suppose we have two periodic functions and we know their Fourier series:
$$ f(t) \sim\sum_{n=-\infty}^{\infty}F_n e^{i n \frac{2 \pi}{T_f} t}\\ g(t) \sim\sum_{n=\infty}^{\infty}G_n e^{i n \frac{2 \pi}{T_g} t} $$
what is the Fourier series coefficients of $h(t)=f(t)\cdot g(t)$?
I know when $T_f = T_g$, it is their convolution
$$ H_n = \sum_{m=-\infty}^{\infty}F_m\cdot G_{n-m}=F_n*G_n $$
But in this case
$$ h(t) \sim \sum_{n=-\infty}^{\infty}{\sum_{m=-\infty}^{\infty}{F_{n}G_{m} e^{i 2 \pi \left( \frac{n}{T_f}+\frac{m}{T_g} \right) t}}} $$
I would like to know the amplitude of each frequency but working this way I have to find several terms and sum them. I have to solve the equation $\frac{n}{T_f}+\frac{m}{T_g}=f_h ,\quad \{n,m\}\in \mathbb{Z}$ then add them all up. It would be better to have $H_n$ and have the sum without the need for solving a equation for each.
Assume that $T_f/T_g$ is rational $p/q$, where $p,q$ are positive integers. Then $T := qT_f=pT_g$ is a common period. The Fourier series of $f$ and $g$ can be written $$\sum_{k \in \mathbb{Z}} a_n e^{i2\pi nt/T} \text{ and } \sum_{k \in \mathbb{Z}} b_n e^{i2\pi nt/T},$$ where $a_{qk}=:F_k$ and $b_{pk}:=G_k$ for every $k \in \mathbb{Z}$, and the other coefficients are $0$. Then you need only to make a convolution of the sequences $(a_n)$ and $(b_n)$ thus defined.