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?



Is the Fourier transform of $\exp(-\|x\|)$ non-negative?
This Fourier transform is positive, supported everywhere, and has polynomial decay. It is the Poisson kernel evaluated at time 1, up to some rescaling.


<http://en.wikipedia.org/wiki/Poisson_kernel>


