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It is well known that a necessary condition for a function in $L_2$ to be a.e. positive is that its fourier transform is positive-definite (in fact, due to Bochner's theorem, this is also a sufficient condition).

Are there any natural necessary conditions, involving the Hermite coefficients of a density in $L_2(R^n, \gamma)$, for this density to be almost-everywhere positive?

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