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For questions about integral transforms, inlcuding the Fourier transform, Laplace transform, Radon transform, Mellin transform, Hankel transform etc.

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What function is a Gaussian integral

Let $g(u,\delta)=E[f(x)]$ where the expectation is over $N(u,\delta^2)$. Is there a characterization what function $g(u,\delta)$ can be produced this way? Is there a procedure solve the inverse proble …
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