Corollary 2

For $f\in A(G)$, let $\Omega_f$ be the set of $\chi \in \widehat{G}$ such that $\mathscr{F}(f)(\chi )\not= 0$. The sets $\Omega_f$ form an open covering of $\widehat{G}$.

This follows from the preceding corollary since, for every $\chi \in \widehat{G}$, the mapping $f\mapsto \mathscr{F}(f)(\chi )$ is a nonzero character of $L^1(G)$.

Recall that the left regular representation $\boldsymbol{\gamma }$ of Stell(G) on $L^2(G)$ (cf. I, p. 125, n$^o13$) is denoted by $\boldsymbol{\gamma }(\varphi )f$ = $\varphi *f$ for $\varphi \in$ Stell(G) and $f\in L^2(G)$.