TITLE: Decay of Fourier coefficients for compact Lie groups
QUESTION [7 upvotes]: 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 $a_\pi$?

REPLY [5 votes]: I believe the Sugiura article is, from MathSciNet, 
(MR0294571 Reviewed) Sugiura, Mitsuo Fourier series of smooth functions on compact Lie groups. Osaka J. Math. 8 (1971), 33–47. 
Apart from deciding on a summation order, there is the more pointed issue of an upper bound for sup-norm in terms of $L^2$ norm of functions in the irreducibles, obviously far less trivial than the bound $|e^{inx}|=1$ for the circle. But it is not hard: up to a measure constant the bound of sup-norm divided by $L^2$ norm is square root of the dimension of the repn (and the repn need not even be irreducible, in fact). I saw this sort of thing proven in Stein-Weiss ("Fourier analysis on Euclidean spaces") and a suitable form of the argument easily generalizes (e.g., see section 6 of my course notes http://www.math.umn.edu/~garrett/m/mfms/notes_2013-14/09_spheres.pdf).