# Decay of Fourier coefficients for compact Lie groups

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 $a_\pi$?

• 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. Jun 27, 2016 at 17:10
• 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 Jun 27, 2016 at 17:36
• 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) Jun 27, 2016 at 17:38
• @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). Jun 27, 2016 at 18:24

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).