Let $f : [0,2\pi] \to \mathbb{R}$ be a continuous convex function on $(0,2\pi)$ which is singular about $0$ and $2\pi$ but finite when evaluated at the boundaries. Assume also that $f$ is symmetric about $\pi$. The discrete Fourier transform of $f$ when sampled at $2\pi i/N$ is then given by $$ X_n := \sum_{i=0}^{N-1}\cos\left(\frac{2\pi n i}{N}\right)f\left(\frac{2\pi i}{N}\right), \quad n = 1,\ldots,N-1. $$
Question: Given the convexity of $f$, does there exist a convex function $G : [0,1] \to \mathbb{R}$ such that $G(n/N) = X_n$? That is, when the DFT is plotted against $n = 1,\ldots,N-1$, does it agree with a convex function. Is the DFT a discrete convex function with respect to $n$?