Parseval identity on a finite group ($\mathbb{E}\overline{f}g={\sum }_{\rho }{d}_{\rho }\mathrm{tr}(\hat{f}(\rho {)}^{\ast }\hat{g}(\rho ))$)

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 $\Vert f{\Vert }_2^2={\sum }_{\rho }{d}_{\rho }\Vert \hat{f}(\rho ){\Vert }_{\mathrm{HS}}^2$. For an abelian group, all $d_{\rho }=1$, and the identity becomes a sum of squared scalar coefficients.