TITLE: Is the Fourier transform of $\exp(-\|x\|)$ non-negative?
QUESTION [7 upvotes]: Is the $n$-dimensional Fourier transform of $\exp(-\|x\|)$ always non-negative, where $\|\cdot\|$ is the Euclidean norm on $\mathbb{R}^n$? What is its support?

REPLY [2 votes]: 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.