Let $f : \mathbb{C} \to \mathbb{C}$ and let the nth Fourier coefficient be defined as:
$$c_n = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(t) e^{-int} dt$$
I just wondered wether there is any way (any sufficient, necessary or sufficient and necessary condition) to determinate wether $c_n >0$ for any sufficiently large $n$ (assuming that we had already proved that, for our function $f$, $c_n \in \mathbb{R}$). I would also be interested in trigonometric coefficients instead of exponential ones.
Reading about it, I found this related question, regarding the positiveness of all of the trigonometric Fourier coefficietns. I also found some interesting information about applying the saddle-point method to this situation, or even expressing $f(z)$ as the product of its modulus and the complex exponential of the argument. However, I have not been able to arrive at any conclusion.