Exercise 32, § 1

Let $G$ be a finite abelian group. We endow $G$ with the Haar measure of total mass 1.

a) Now prove that if $f$ is non zero, then

Card(Supp($f$)) Card(Supp($\widehat{f}$))$\geqslant$ Card(G)

(“Uncertainty principle”; note that

1

Card(G)

$x\in G$

and apply the Cauchy-Schwarz inequality and the Plancherel formula.)

b) Determine the cases of equality.