= Normalized Fourier analysis on a finite abelian group
For a finite abelian group $G$, use the normalized average $\mathbb E_x=|G|^{-1}\sum_x$, normalized convolution
$$
(f*g)(x)=\mathbb E_y f(x-y)g(y),
$$
and Fourier transform $\widehat f(\gamma)=\mathbb E_xf(x)\overline{\gamma(x)}$. Then Parseval's identity and the convolution identity take the forms
$$
\langle f,g\rangle=\sum_{\gamma\in\widehat G}\widehat f(\gamma)\overline{\widehat g(\gamma)},
\qquad
\widehat{f*g}(\gamma)=\widehat f(\gamma)\widehat g(\gamma).
$$
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