Exercise 68, § 1

We denote additively the group law of G. Let Γ be a discrete subgroup of G such that $G/\Gamma$ is compact. Let $\Lambda = \Gamma^{\bot}$ be the orthogonal of Γ in $\widehat{G}$; since $\widehat{G}/\Gamma$ identifies with Λ, it is a discrete subgroup of $\widehat{G}$, and $\widehat{G}/\Lambda$ is compact (since its dual identifies with Γ).

We suppose that the Haar measure on G is chosen in such a way that the quotient of the dual measure $d\chi$ on $\widehat{G}$ by the counting measure on Λ is the normalized Haar measure on $\widehat{G}/\Lambda$ (INT, VII, § 2, n$^o2$, def. 1).

a) There exists a measurable subset Ω of $\widehat{G}$ which meets each class of G modulo Λ at one and only one point; the measure of Ω is equal to 1.

b) For $x\in G$, we put

The function $\varphi$ is continuous; it satisfies $\varphi (0) = 1$ and $\varphi (\gamma ) = 0$ for every $\gamma \in \Gamma -\{0\}$.

c) Exercise prove that $\varphi$ belongs to $L^2(G)$ and that $\|\varphi \|_2= 1$. For every $\gamma \in$ Γ $-\{e\}$, one has

d) Let $f\in L^2(G)$ such that $\mathscr{F}(f)$ is null almost everywhere outside Ω. Exercise prove that

in $L^2(G)$, where $\boldsymbol{\delta }_{\gamma}(\varphi )(x) =\varphi (x-\gamma )$. Moreover, one has

e) There exists a unique continuous function on G which coincides almost everywhere with $f$. If $f$ is continuous, then

in $\mathscr{C}(G)$.

f) Specialize to the case where $G =\mathbf{R}$ and $\Lambda =h\mathbf{Z}$ for a certain $h >0$ by taking $\Omega = [-h, h[$.