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 mapping from $G$ to the set of complex numbers.

Then for all $x \in G$:


 * $\ds \map \eta x = \frac 1 {\map \phi q} \sum_{\chi \mathop \in G^*} \innerprod \eta \chi_G \map \chi x$

where:


 * $\ds \innerprod \eta \chi_G = \sum_{x \mathop \in G} \map \eta x \map {\overline \chi} x$