Exercise 6, § 4

Let $G$ be a compact group and $\varrho$ an irreducible representation of $G$. There exists a central function $f\in \Theta (G^{\sharp})$ such that

where the integers $c(\pi )$ satisfy

(i) $c(\pi )\geqslant 0$ for all $\pi \in \widehat{G}$;

(ii) $c(1)\leqslant c(\varrho )$, and the inequality is strict unless possibly if $\varrho$ is of dimension 1 and satisfies $\varrho^2= 1$;

(iii) $\mathscr{R}(f)\geqslant 0$.