Let $G$ be a compact topological group. By the Peter-Weyl theorem, the complex Hilbert space $L^2(G)$ is the Hilbert space direct sum of the spaces of matrix coefficients of all the irreducible representations of $G$. When $G$ is commutative, these irreducible representations are one-dimensional, and the Peter-Weyl theorem just says that the unitary characters of $G$ form an orthonormal basis of $L^2(G)$.
For example, when $G = \mathbb R/\mathbb Z$, the functions $f_n(x) = e^{2\pi i nx}$ are an orthonormal basis of $L^2(G)$. Thus if $f \in L^2(G)$, then there are unique complex numbers $c_n$, with $\sum\limits_n |c_n|^2 < \infty$, which gives $f$ its Fourier expansion:
$$f = \sum\limits_{n\in \mathbb Z} c_n f_n$$ where the sum on the right hand side converges in the $L^2$-norm to $f$. It is a much deeper theorem that when $f$ is continuous, the right hand side also converges pointwise to $f$ almost everywhere.
Is the analogue of this deep theorem known for other compact groups? That is, suppose we take an orthonormal basis $f_i : i \in I$ of $L^2(G)$ via matrix coefficients of irreducible representations as in the Peter-Weyl theorem, so that any $f \in L^2(G)$ has a "Fourier expansion" for uniquely determined $c_i \in \mathbb C$,
$$f = \sum\limits_{i \in I} c_i f_i$$ so the right hand side converges to $f$ in the $L^2$-norm. Suppose that $f$ is continuous. Then, do we know that the right hand side converges pointwise to $f$ almost everywhere?
If this is not known in general, is it known for, say, $G = \mathbb A_k/k$ for $k$ a global field?
