Exercise 19, § 4

Let G be a compact group endowed with its normalized Haar measure $\mu$ and let $f\in \mathscr{C}(G)$ be a continuous function on G. There exists an irreducible representation $\pi \in \widehat{G}$ such that $f=$ dim($\pi$ )$^{-1}\chi_{\pi}$ if and only if

for all $(g, h)\in G^2$.