Example

Let Q be a positive definite quadratic form on $\mathbf{R}^n$. The function defined by $\varphi (x) =$ exp($-\pi Q(x)$) belong to $\mathscr{S}(\mathbf{R}^n)$. There exists $\sigma \in$ GL($n,\mathbf{R}$) such that $Q(x) =\|\sigma (x)\|^2$ for every $x\in \mathbf{R}^n$. The Fourier transform of $\varphi$ is given for every $y\in \mathbf{R}^n$ by

$\widehat{\varphi}(y) =|$det($1\sigma$ )$|$ exp($-\pi Q^*(y)$)

where $Q^*(y) =\|^t\sigma^{-1}(y)\|^2($cf. INT, IX, §6, n$^o$ 4–5 and exercise 1, c) of II, p. 262).