Given a Fourier series $g(x)=\sum_{m=-M}^{M}\,a_m e^{i m x}$ such that $|g(x)|\leq 1$ for all $x\in [-\pi,\pi]$. Is it always possible to find a "complementary" Fourier series $h(x)$ of the same order $M$ such that $|g(x)|^2+|h(x)|^2=1$ for all $x\in [-\pi,\pi]$?
I know that this is true for some cases, but I wish I could prove it for an arbitrary Fourier series $g(x)$.
I don't know why we would want to consider the sum of the norms. Usually we want to consider $|g(x)+h(x)|^2$ for example to see if we can get it arbitrarily small.
Fourier series can arbitrarily well in squared integral sense approximate all functions in L2 on this interval. You can even find $h$ so that $g(x)+h(x) = f(x)$ for a given $f$ in L2. ( This only holds if $M$ is allowed to get arbitrarily big. )
But in particular for sines and cosines which are the basis functions in the Fourier series we have trigonometric unity for every frequency which maybe is what you have in mind ?
$$\sin(nt)^2 + \cos(nt)^2 = 1, \forall n$$
So you could just make all terms cancel except of one sine cosine pair.
You could choose which one.
We can realize this is related to the complex exponential formulation of Fourier Series by considering Euler's formula $$\exp(ti) = \cos(t)+i\sin(t)$$
For the particular $|g(t)|^2+|h(t)|^2 = 1$
I suppose there could be utility in ensuring sum of power draw in two electrical circuits following a smooth curve, optimally a constant one.
Sorry I can't think of a proof.