= Parseval identity on a finite group
{c}
{title2=$\mathbb E\overline fg=\sum_\rho d_\rho\operatorname{tr}(\widehat f(\rho)^*\widehat g(\rho))$}
For the <Fourier transform on a finite group>, the displayed identity uses uniform <expectation> in the original function space and a weighted <Hilbert-Schmidt inner product> in the matrix components. In particular $\|f\|_2^2=\sum_\rho d_\rho\|\widehat f(\rho)\|_{\mathrm{HS}}^2$. For an <abelian group>, all $d_\rho=1$, and the identity becomes a sum of squared scalar coefficients.
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