= Convolution theorem on a finite group
{title2=$\widehat{f*g}(\rho)=\widehat f(\rho)\widehat g(\rho)$}
The <normalized convolution on a finite group> and the <Fourier transform on a finite group> convention with $\rho(x)$ satisfy the displayed identity. Substitute $x=yz$ in the defining <expectation> and use $\rho(yz)=\rho(y)\rho(z)$. With the convention using $\rho(x)^*$ instead, the scalar convolution has transform $\widehat g(\rho)\widehat f(\rho)$; multiplication order matters for noncommutative <groups>.
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