Fourier transform of a function over finite group

Question

Let $G$ be finite abelian group and $\hat G$ be its character group.

The Fourier transform of a function $f:G \to \mathbb C$, is the function $\hat{f}:\hat{G}\to \mathbb C$ defined by $\hat{f}(\chi)=\sum_{a\in G}f(a)\chi(-a)$.

I need hint proving that $\hat{\hat{f}}(a)=|G|f(-a)$.

Answer 1

Hint

$$\hat{\hat{f}}(a)=\sum_{\chi\in \hat{G}}\hat{f}(\chi)\chi(-a)=\sum_{\chi\in \hat{G}}\sum_{b\in G}f(b)\chi(-b)\chi(-a)=\sum_{ b \in G}f(b)\left(\sum_{\chi\in \hat{G}} \chi(-a-b)\right) \,.$$

For $c \in G$ what is

$$\sum_{\chi\in \hat{G}} \chi(c) ?$$

Answer 2

The very first step is to unpackage the definitions and write stuff down:

$$\widehat{\widehat{f}}(a)=\sum_{\chi\in\widehat{G}}\hat{f}(\chi)\bar{\chi}(a)=\sum_{\chi\in\widehat{G}}\left[\sum_{g\in G}f(g)\chi(-g)\right]\bar{\chi}(a)=\cdots$$

Have you done this yet? After writing things down, you want to rearrange stuff. The most obvious thing to rearrange is the order of the summations. Then you want to invoke any character-theoretic properties you have at your disposal that look relevant. Like, say, orthogonality relations..