= Fourier coefficient on a finite abelian group
{c}
{title2=$\widehat f(\chi)=\mathbb E_xf(x)\overline{\chi(x)}$}
= Fourier coefficients on a finite abelian group
{c}
{synonym}
= Finite abelian Fourier coefficient
{synonym}
For a scalar function on a <finite abelian group>, its coefficient at an <additive character> $\chi$ is the displayed uniform <expectation>. On $\mathbb F_3^d$, write $\chi_\xi(x)=e^{2\pi i\xi\cdot x/3}$. The coefficient at the constant character is the mean of $f$, and the <Parseval identity on a finite group> becomes $\sum_\chi|\widehat f(\chi)|^2=\mathbb E|f|^2$ after relabelling characters to match the transform convention.
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