Remark 1

The Fourier transform of $(u_{\pi})_{\pi\in\widehat{G}}\in F^2(\widehat{G})$ is therefore the class in $L^2(G)$ of the series

$g\mapsto \sum_{\pi\in\widehat{G}}\langle \pi (g)|u_{\pi}\rangle =\sum_{\pi\in\widehat{G}}$ dim($\pi$)Tr($u_{\pi}\circ \pi (g)^{-1}$).