Discrete Fourier Transform on Abelian Group

Theorem
Let $G$ be a finite abelian group.

Let $G^*$ be the dual group of characters $G \to \C^\times$.

Let $\eta : G \to \C$ be a function.

Then for all $x \in G$,


 * $\displaystyle \eta(x) = \frac{1}{\phi(q)} \sum_{\chi \in G^*} \langle \eta, \chi \rangle_G \chi(x)$

where


 * $\displaystyle \langle \eta, \chi \rangle_G = \sum_{x \in G} \eta(x) \overline{\chi}(x)$

Proof
We have

This proves the result.