Let $G$ be a compact, second countable, Hausdorff topological group with the normalized Haar measure $\mu$. From Peter-Weyl's theorem we now that for any $f\in \mathrm{L}^2(G)$ the Fourier series of $f$: $$ \sum_{\rho}d_\rho\langle \hat{f}(\rho),\rho(g)\rangle_{HS} $$ converges in $\mathrm{L}^2(G,\mu)$ to $f$. My first question is:

- Why do people usually write $f(g)=\sum_{\rho}d_\rho\langle \hat{f}(\rho),\rho(g)\rangle_{HS}$? This notation suggest that the series converges pointwise to $f(g)$ which is definitely wrong. Is it a conventional notation or there is reason behind it?
- What can we say about the pointwise convergent points of the Fourier series?
- Let $G$ be a profinite group. Is it true that the Fourier series converges pointwise to $f$?