Exercise 55, § 1

We retain the notations of the preceding exercise. We equip $\widetilde{G}$ with the normalized Haar measure $\nu$. Let $f$ be a function almost periodic on G and $\widetilde{f}$ its extension to $\widetilde{G}$. We set

a) Let $\chi \in \widehat{G}$. The function $\chi f$ is almost periodic. We then define $\mathscr{P}(f)(\chi ) = A(\chi f)$. We have

and in particular $A(\chi f) = 0$ except for a countable set of $\chi$.

b) Let $f$ be a function almost periodic on $\mathbf{R}$. For every $y\in \mathbf{R}$, we have

1 $\int^T$

$\mathscr{P}(f)(y) =$ lim $f(x)e^{-2i\pi xy}dx$.

c) Let Λ be a finite subset of $\mathbf{R}$. The function defined on $\mathbf{R}$ by

is almost periodic on $\mathbf{R}$; determine in terms of Λ the set of $y\in \mathbf{R}$ such that $\mathscr{P}(f)(y)\not= 0$.