Let $f : [0,2\pi] \to \mathbb{R}$ be some function. Then 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:What conditions are sufficient on $f$ such that there exists a convex function $G : [0,1] \to \mathbb{R}$ with $G(n/N) = X_n$? That is, when is the DFT a discrete convex function with respect to $n$?

- In the references post above it was shown that $f(x) = |\pi-x|$ defies this claim.

**Notes**: This question is related to the recent post: Convexity of discrete Fourier transform