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Im working in generating 1D isotropic random mediums and I arrived that to predefine the covariance of the medium, it needs to satisfy

$\hat C (\xi) \geq 0 \quad \forall \xi \in \mathbb{R}$

where $C(x)$ is the covariance of any 2 points at distance $x$.

Since $C(x)$ is depending on the distance, it is an even function and thus the Fourier transform is a real valued function. My question is if there is any characterization for even functions that have non-negative Fourier transforms.

Ethan Bolker
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pancho
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  • the convolution by $C(x)$ operator is positive semi-definite (as a self-adjoint matrix whose eigenvalues are $\ge 0$) – reuns Feb 23 '16 at 15:33
  • hence $u \to \langle u, u \ast (C+\epsilon \delta) \rangle$ is a (Hilbert space) norm whenever $\epsilon > 0$ – reuns Feb 23 '16 at 15:42
  • Thats right, thank you ! but I'm looking for a more "visual" characterization. Something that can easily allow me to check if an input covariance is allowed. – pancho Feb 23 '16 at 17:17

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Actually I only wanted to comment.. (but I lack reputation)

If $C(X)$ is also positive, consider https://arxiv.org/abs/math-ph/0504015

alain
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