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