Discrete Fourier Transform on Abelian Group

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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 $x \in G$:

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

where:

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


Proof

\(\displaystyle \frac 1 {\phi \left({q}\right)} \sum_{\chi \mathop \in G^*} \langle \eta, \chi \rangle_G \chi \left({y}\right)\) \(=\) \(\displaystyle \frac 1 {\phi \left({q}\right)} \sum_{\chi \mathop \in G^*} \sum_{x \mathop \in G} \eta \left({x}\right) \overline \chi \left({x}\right) \chi \left({y}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {\phi \left({q}\right)} \sum_{x \mathop \in G} \eta(x) \sum_{\chi \mathop \in G^*} \overline \chi \left({x}\right) \chi \left({y}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {\phi \left({q}\right)} \eta \left({y}\right) \phi \left({y}\right)\) Orthogonality Relations for Characters
\(\displaystyle \) \(=\) \(\displaystyle \eta \left({y}\right)\)

$\blacksquare$