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Sufficient situations for the convexity of the discrete Fourier transforms

Let $f : [0,2pi] to mathbb{R}$ breathe some duty. Then the *discrete Fourier rework* of $f$ when sampled at $2pi i/N$ is then given by

$$

X_n := sum_{i=0}^{N-1}cosleft(frac{2pi n i}{N}privilege)fleft(frac{2pi i}{N}privilege), quad n = 1,ldots,N-1.

$$

Question:What situations are adequate on $f$ such that there exists a convex duty $G : [0,1] to mathbb{R}$ with $G(n/N) = X_n$? That is, when is the DFT a discrete convex duty with respect to $n$?

- In the references publish above it was proven that $f(x) = |pi-x|$ defies this pretense.

**Notes**: This query is said to the latest publish: Convexity of discrete Fourier rework

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