This question spanned from a previous interesting one.
Let $k$ be a real number greater than $2$ and
$$\varphi_k(\xi) = \int_{0}^{+\infty}\cos(\xi x) e^{-x^k}\,dx $$
the Fourier cosine transform of a function in the Schwartz space.
Is is possible to use the Fejér-Riesz theorem or some variation of it,
to prove that $\varphi_k(\xi)<0$ for some $\xi\in\mathbb{R}^+$?