Prove the inverse Fourier transform of Gaussian kernel is Gaussian distribution (by Bochner's)

How do I prove$$p(w) = \int_{\mathbb{R}^D} e^{-jw^T \delta} k(\delta) d\delta $$(i.e. $p(w)$ is the inverse Fourier transform of $k(\delta)$) where $p(w)$ is the multivariate Gaussian distribution$$p(w) = (2 \pi /\sigma)^{-D/2} e^{-\sigma w^Tw/2}$$and $k(\delta)$ is the Gaussian kernel$$k(\delta) = e^{-\delta^T\delta/2\sigma}?$$This paper guarantees it but I don't really see how this is proved by Bochner's Theorem.