Let $n$ be an even integer. Find the Fourier-transform of the function $f:C_{nv}\rar\C$, $f=\d_{\s}$, i.e. $f(\s)=1$ and $f(x)=0$ for all $x\neq\s$.

$\wh f(\chi_1)=1/2n$, $\wh f(\chi_2)=-1/2n$, $\wh f(\chi_3)=1/2n$, $\wh f(\chi_4)=-1/2n$ and for $j=1,\ldots,n/2-1$: $$ \wh f(\psi_j)=\frac1{2n} \left(\begin{array}{cc} 0&1\\ 1&0 \end{array}\right)~. $$ Thus the Fourier inversion formula yields: $$ f(x)=\frac1{2n}\Big(\chi_1(x)-\chi_2(x)+\chi_3(x)-\chi_4(x)\Big) +\frac1n\sum_{j=1}^{n/2-1}\tr\left(\left(\begin{array}{cc} 0&1\\ 1&0 \end{array}\right)\Psi_j(x)\right) $$