I am trying to find a proof for why or why not this is possible:
I would like to take a generic sine wave, $y = \sin(x)$ and bound it between $\pm 0.5$, only through the use of third order harmonics. In other words, I want to know if this is ever true: $$-0.5\leq\sin(x) + a_3\sin(3x +\phi_3) + a_9\sin(9x+\phi_9) +\cdots +a_{3^n}\sin(3^nx+\phi_{3^n}) \leq0.5$$
I know that a solution could be found piecewise, but could a solution exist with constant $a_k$ and $\phi_k$?
Thanks for the help.