Proposition 21

The restriction of the Fourier transform to $\mathscr{S}(\mathbf{R}^n)$ is an automorphism of topological vector spaces whose inverse is the restriction of the Fourier cotransform.

Let $\Lambda \subset \mathbf{R}^n$ be a lattice (TG, VII, p. 4), and let $\Lambda^*\subset \mathbf{R}^n$ be the associated lattice (TG, VII, p. 6), also sometimes called the dual lattice.

The covolume of the lattice Λ is, by definition, the measure of $\mathbf{R}^n/\Lambda$ for the Haar measure induced by Lebesgue measure on $\mathbf{R}^n($cf. INT, VIII, §5, n$^o5$, example), and is denoted by V(Λ). For every function $f\in$ $\mathscr{S}(\mathbf{R}^n)$ and every $y\in \mathbf{R}^n$, one has the Poisson formula

1

$\sum f(x+y) =\sum\widehat{f}(z)$ exp(2$i\pi y\cdot z$).

V(Λ) \*

$x\in \Lambda z\in \Lambda$