Fourier series of the fractional part

Question

What is the Fourier series for $\{a\}\{b\}$, i.e. the product of the fractional parts of $a$ and $b$. I know what the Fourier series looks like for a single value of either $a$ or $b$, but I want to know what it is when the two are multiplied together.

Answer 1

If $0 \leq \{a\} < 1$ and $0 \leq \{b\} < 1$, then $0 \leq \{a\}\{b\} < 1$. So the Fourier series of the product would look just like the Fourier series of $\{c\}$ of some real number $c$.