# Sufficient situations for the convexity of the discrete Fourier transforms Answer

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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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