FOURIER TRANSFORM II

Because of analysis, everybody knows the Fourier transform

$$\hat f(x) =\int_{{\bf R}} f(t) e^{itx} dt$$

It has an algebraic analog

$$\sum_{x\in G} f(x) \chi(x)=\hat f(\chi)$$

A character $\chi$ of $G$ is an homorphism from the group$G$ into ${{\bf C}}^\times$

    orthogonality relation

$$\sum_{s\in S} \chi(s)=\begin{cases}\vert S\vert ,&\text{if $\chi\bot S$ ;}\\0,&\text{else.}\\\end{cases}$$
 

    Character sums method.

Let $f$ be a mapping from $X$ into $G$, the number $N(f,c)$ of solutions of $f(x)=c$ in $X$ is given by

$$\begin{split}N(f,c) &= \frac 1{\vert G\vert} \sum_{\chi}\sum......{\vert G\vert} \sum_{\chi} S(f,\chi) \bar\chi(c)\\\end{split}$$
 

    Trivialisations.
Let $f$ and $g$ the product of convolution of $f$ by $g$

$$\begin{align*}f\ast g(z) &=\sum_{x+y=z} f(x)g(y),\\f\times g(z) &=\sum_{x-y=z} f(x)g(y)^{\ast}\end{align*}$$

the fourier transform is an homomorphism
from ${\bf C}[G]$ into${\bf C}^{\hat G}$

$$\begin{align*}\widehat{f\ast g}(\chi) &= \hat f(\chi)\hat g(\chi),\\\widehat{f\times g} &= \hat f(\chi){\hat g(\chi)}^{\ast}\\\end{align*}$$

    Poisson Formula.
Let $\Gamma$ be a subgroup of $\hat G$

$$\frac {1}{\vert \Gamma\vert} \sum_{\chi\in\Gamma} \hat f(\chi) \bar\chi(z)= \sum_{s\bot \Gamma} f(s+z)$$
 

simple but useful !

• McWilliams formula
• McEliece weight formula
• Multiplier theorem of Hall -Turyn
• degree of bent functions
• etc...