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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 $a_\pi$?

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    $\begingroup$ It is of course not true that $f:S^1\to\mathbb C$ will be AC if $\widehat{f}(n)=O(1/n)$. This condition just says that $f'$ has Fourier coefficients $O(1)$, which is not enough to ensure that $f'\in L^1$. Since there's no (known) easy description of the Fourier coefficients of $L^1$ functions, there won't be such a characterization of $AC$ either. $\endgroup$
    – Christian Remling
    Commented Jun 27, 2016 at 17:10
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    $\begingroup$ Also, when you write your infinite sum, what is your summation process? Even for G a torus of dimension at least two, different ways of summing the Fourier series are not guaranteed to give the same answer $\endgroup$
    – Yemon Choi
    Commented Jun 27, 2016 at 17:36
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    $\begingroup$ It is however true that there should be results of the form "if the coefficients satisfy this decay condition then the function is C^k" and "if the function is C^k then the coefficients satisfy this other decay condition" which for many purposes are good enough. IIRC there is a paper of Sugiura which has some estimates of this form (too busy watching my country's Blue Screen Of Death to dig up the reference, I'm afraid) $\endgroup$
    – Yemon Choi
    Commented Jun 27, 2016 at 17:38
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    $\begingroup$ @YemonChoi We can think of $\widehat G$ as living inside $\mathfrak{t}^\vee$, where $\mathfrak{t}=\mathrm{Lie}(T)$ is the Lie algebra of the maximal torus. We can order the sum $\sum_\pi a_\pi tr(\pi)$ by looking at those $\pi$ which live in increasingly large balls (any choice of metric on $\mathfrak{t}^\vee$ will do). $\endgroup$
    – Daniel Miller
    Commented Jun 27, 2016 at 18:24

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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).

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    $\begingroup$ Yes, that's the article I was thinking of (which is in my depressingly large folder of Things I Skimmed Once And Filed Away To Read When I Have Time) $\endgroup$
    – Yemon Choi
    Commented Jun 27, 2016 at 20:35

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