Deduce the existence of a real $\nu_{\sigma}$ such that, for any real $\xi$ and any real $t > 0$, $$\hat{f}(t, \xi) = \nu_{\sigma} \exp\left(-2\pi^{2}\left(\sigma^{2}+2t\right) \xi^{2}\right)$$
Deduce the existence of a real $\nu_{\sigma}$ such that, for any real $\xi$ and any real $t > 0$,
$$\hat{f}(t, \xi) = \nu_{\sigma} \exp\left(-2\pi^{2}\left(\sigma^{2}+2t\right) \xi^{2}\right)$$