Let $G$ be a compact Lie group, $G^\natural$ the space of conjugacy classes in $G$ with the natural pushforward of $G$'s Haar measure. Let $f\in L^2(G^\natural)$. Then the Peter–Weyl Theorem tells us that there is a unique sequence $\{a_\pi\}_{\pi\in \widehat G}$ such that $$ f = \sum_\pi a_\pi \mathrm{tr}(\pi) $$ in $L^2(G^\natural)$. This doesn't guarantee that $$ f(x) = \sum_\pi a_\pi \mathrm{tr}(\pi(x)) $$ for all $x\in G^\natural$.

If $G=S^1$, then $\widehat G=\mathbf{Z}$, and there are nice equivalences of the form ``$f$ is absolutely continuous iff $\widehat f(n) = O(1/n)$."

Is there a similar characterization of (absolutely) continuous (/ $C^k$ and/or smooth) functions on general $G^\natural$ in terms of the decay of their $a_\pi$?