Is the Fourier transform of $exp(-|x|)$ non-negative?

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.