I am trying to find the max value over $t$ of
$$\sum_{k} a_k \sin (kt),$$
where $k$ is positive and odd.
If the exact max cannot be found in terms of the real coefficients $a_k$, an upper bound for the sum would also help.
Finding the derivative with respect to $t$ and forcing it to zero did not help, as finding all the roots of
$$\sum_k \frac{a_k}{k} \sin (k t)$$
seems to be at least just as difficult.
The coefficients $a_k$ are bounded with
$$|a_k|<\frac{1}{k},$$ but this bound is not tight, so I would prefer not to use it.
Any ideas where I should look?