Solution
= Solution
Integration by parts gives $\tilde g_\sigma\prime(k)=-\sigma^2k\tilde g_\sigma(k)$ and $\tilde g_\sigma(0)=1$, so $\tilde g_\sigma=e^{-\sigma^2k^2/2}$. The convolution theorem says $\widetilde{f*g}=\tilde f\tilde g$; multiplying the two Gaussian transforms and inverting yields $g_{\sqrt{\sigma_1^2+\sigma_2^2}}$.