= Fourier transform on a finite group
{c}
{title2=$\widehat f(\rho)=\mathbb E_xf(x)\rho(x)$}
For a scalar function on a <finite group>, one normalized transform convention assigns the matrix $\mathbb E_xf(x)\rho(x)$ to each chosen <unitary irreducible representation>. This map is a weighted <Hilbert space> isomorphism by the <Parseval identity on a finite group>. Another common convention uses $\rho(x)^*$; the corresponding <convolution theorem on a finite group> then reverses the matrix product order for $(f*g)(x)=\mathbb E_yf(y)g(y^{-1}x)$.
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