Is the $n$dimensional Fourier transform of $\exp(\x\)$ always nonnegative, where $\\cdot\$ is the Euclidean norm on $\mathbb{R}^n$? 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. 


These questions are closely related to the socalled 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 integrableotherwise the 'universal' Central Limit Theorem would hold. The cauchy distribution is only weakly integrable. 

