Show that, for all $\xi \in \mathbb{R}$, there exists a real $K(\xi)$ such that for all $t \in \mathbb{R}_{+}^{*}$, $\hat{f}(t, \xi) = K(\xi) \exp\left(-4\pi^{2} \xi^{2} t\right)$.
Show that, for all $\xi \in \mathbb{R}$, there exists a real $K(\xi)$ such that for all $t \in \mathbb{R}_{+}^{*}$, $\hat{f}(t, \xi) = K(\xi) \exp\left(-4\pi^{2} \xi^{2} t\right)$.