We consider the function $\theta : \mathbb{R} \rightarrow \mathbb{C}$ defined by $\theta(x) = \exp(-\pi x^{2})$, for $x \in \mathbb{R}$.
I.E.1) Justify that $\theta$ belongs to $\mathcal{S}$ and that $\mathcal{F}(\theta)$ is a solution of the differential equation
$$\forall \xi \in \mathbb{R}, \quad y'(\xi) = -2\pi\xi\, y(\xi)$$
I.E.2) Establish that $\mathcal{F}(\theta) = \theta$.
We will admit that $\int_{-\infty}^{+\infty} \theta(x) \mathrm{d}x = 1$.