Fourier coefficient on a finite abelian group ($\hat{f}(\chi )={\mathbb{E}}_{x}f(x)\overline{\chi (x)}$)

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 ${\sum }_{\chi }|\hat{f}(\chi ){|}^2=\mathbb{E}|f{|}^2$ after relabelling characters to match the transform convention.