Given a sequence of real points $a_k$ for $k = 0,\ldots,N-1$ such that $a_{k} = a_{N-k}$ for $k = 0,\ldots,N-1$ (assuming $a_N = a_0$), what conditions on $(a_k)_{k=0}^{N-1}$ would be sufficient for the discrete Fourier transform of this sequence to be positive, i.e. $$ X_n := \sum_{l=0}^{N-1}\cos\left(\frac{2\pi n l}{N}\right)a_l \geq 0, \quad n = 1,\ldots,N-1? $$
Alternatively, say $a_k = f(x_k)$ are sample points of a convex function, would this be sufficient for positivity? I'm not sure how to show this. Any help would be appreciated.