Is the n-dimensional Fourier transform of exp(-||x||) always non-negative, where ||.|| is the Euclidean norm on R^n ? What is its support?
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This Fourier transform is positive, supported everywhere, and has polynomial decay. It is the Poisson kernel evaluated at time 1, up to some rescaling. |
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Thanks, Josh. (I wanted to know too.) Do you know the answer when ||.|| is the p-norm for arbitrary p in [1, 2]? |
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These questions are closely related to the so-called stable distributions. In particular, the cauchy distribution on the real line has the characteristic function e^{-|x|}. Go to the wikipedia page, and in the definition section set: mu=0 (this is the drift parameter) alpha=0 (this is the skewness parameter) To get the same thing in higher dimensions, take independent copies in each coordinate. Take note: These distributions are not square integrable--otherwise the 'universal' Central Limit Theorem would hold. The cauchy distribution is only weakly integrable. |
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