= Fourier analysis on a finite group
{c}
{title2=$\widehat f(\rho)=\mathbb E_x f(x)\rho(x)$}
= Non-abelian Fourier analysis
{synonym}
For a <finite group>, choose one <unitary irreducible representation> $\rho$ of degree $d_\rho$ from each equivalence class. One consistent normalized <Fourier transform on a finite group> convention is $\widehat f(\rho)=\mathbb E_xf(x)\rho(x)$. The <Schur orthogonality relations> give
$$
f(x)=\sum_\rho d_\rho\operatorname{tr}(\widehat f(\rho)\rho(x)^*),
\qquad
\mathbb E_x|f(x)|^2=\sum_\rho d_\rho\|\widehat f(\rho)\|_{\mathrm{HS}}^2.
$$
The <normalized convolution on a finite group> satisfies $\widehat{f*g}(\rho)=\widehat f(\rho)\widehat g(\rho)$ in this convention. Using $\rho(x)^*$ in the transform instead reverses that product order for scalar functions with the same <normalized convolution on a finite group> convention. The <Fourier transform on a finite group> has matrix-valued components even when the original function is scalar-valued.
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