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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.

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user3141592
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  • I don't think there is any simpler condition than "$f$ must be the sum of a trigonometric polynomial and a function with positive Fourier coefficients", the latter being described by the linked post. –  Apr 30 '18 at 23:06
  • It may be simple to express, but it is definitively such a difficult way to check wether a given a funcion $f(z)$ satisfies that condition – user3141592 May 02 '18 at 09:32
  • I'm a little confused about your definition of $c_n$. Is this a path integral (as $dz$ suggests) or an integral on $\mathbb{T}$ (as the bounds of $-\pi$ and $\pi$ suggest)? –  Jun 21 '18 at 08:09
  • @Ryan Corrected notation to prevent misunderstandings – user3141592 Jun 21 '18 at 23:56

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