Let $f(\omega) = \sum_{k = 0}^{n-1} w_k \cos(k \omega)$ with $\omega \in [-\pi, \pi]$ be a finite and real Fourier series.
Is it possible to characterize for which coefficients $w_k \in \mathbb{R}$ this series is non-negative (i.e. $f(\omega) \geq 0$)?
Edit: Is is also possible to characterize for which coefficients $w_k \in \mathbb{R}$ this series is non-negative on the discrete domain $\{\omega_0, \cdots, \omega_{m-1}\} \subset [-\pi, \pi]$?
$w_k=\int_{-\pi}^{\pi}e^{ik x}\mu(dx)$ for some positive measure $\mu$ if and only if $$\sum_{k,\ell}w_{k-\ell}\overline{c_k}c_{\ell}\geq 0$$ for all $(c_k)_{1}^n$ and all $n$. This is the Bochner theorem for the group $\mathbb{Z}.$ Have a look to Rudin, Real and complex analysis.
Edit. $f(x)=\sum_{k=-n}^nw_k e^{ikx}$ with $w_{-k}=\overline{w_k}$ implies that $w_k=\int_{-\pi}^{\pi}e^{-ikx}f(x)dx$ and therefore if $f>0$ we have $$\sum_{k,\ell}w_{k-\ell}\overline{c_k}c_{\ell}=\int_{-\pi}^{\pi}f(x)|\sum_{k=-n}^nc_ke^{ik x}|^2dx\geq 0$$ which is saying that the Hermitian matrix $(w_{k-\ell})$ is semi positive definite (this matrix is symmetric if you insist for taking $w_k$ real). The Bochner theorem says that conversely if the Hermitian matrix $(w_{k-\ell})$ is semi positive definite, then the trigonometric polynomial $f$ is positive. May be it is better to read the first chapter of Rudin Fourier Analysis on Groups, 1963.