Fouries series of $\sin{\sum_n a_n \sin{n\theta}}$

Question

What's the Fourier series for $\sin({\sum_n a_n \sin{n\theta}})$?

Answer 1

There is no particular formula for the Fourier series this kind of function, other than the definition of Fourier series. While Fourier coefficients are nicely transformed under linear maps, the relation between the Fourier series of $f$ and $\sin f$ is completely opaque.

For example, take $\sin \sin \theta$. Its Fourier series begins with $$ 2J_1(0)\sin\theta + (14J_1(1)-8J_0(1))\sin 3\theta +(626 J_1(1)-380 J_0(1))\sin 5\theta +(73534 J_1(1)-42288J_0(1)) \sin7\theta +\dots $$ a bunch of very nice integers of course, but not something you could get directly from $a_1=1$. To say nothing of the Bessel functions appearing in the coefficients above.