There is a well-known relation known as Jacobi-Anger expansion: $$ e^{ix \sin\theta} = \sum_{n=-\infty}^{\infty}J_{n}(x)e^{in \theta}, $$
where $J_{n}$ are the Bessel functions of the first kind.
Do you know any other similar expansions for some periodic function $f(t)$ besides sinsuoidal function, such that:
$$ e^{if(t)}=\sum_{k =-\infty}^{\infty} a_k e^{ik \theta}. $$