For a given function $f\in C(G)$ on a compact group $G$ its Fourier transform is defined as the family of operators
$$
\widehat{f}_\sigma=\int_Gf(t)\cdot\sigma(t^{-1}) \ \text{d}\ t,\quad \widehat{f}_\sigma:X_\sigma\to X_\sigma
$$
where $\sigma:G\to B(X_\sigma)$ runs over the dual object $\widehat{G}$ and $\int...\text{d}\ t$ means the integral with respect to the normed Haar measure (i.e. $\int 1\ \text{d}\ t=1$). It is known that in the space $L_2(G)$ the following equality holds:
$$
f(t)=\sum_{\sigma\in\widehat{G}}\dim X_\sigma\cdot\text{tr}\Big(\widehat{f}_\sigma\circ\sigma(t)\Big),\quad t\in G.
$$
I wonder if this can be interpreted as an equality in $C(G)$ in some specific sense (or as a pointwise equality)? As an example, in the case $G=\mathbb{T}$ the Fejér theorem states that the arithmetical means of the partial sums of the series converge to $f$ in $C(G)$ (and hence pointwisely). Is it possible that someting similar is true for all compact groups?
P.S. I asked this also in MathOverflow.
