The spectrum recovers the selector
On this page · Permalink · Source line 74
The Fourier transform returns exactly the four labels selected by coprimality. Use the negative-sign, unnormalized discrete Fourier transform
\[
\widehat x(k)=\sum_{n=0}^{11}x(n)e^{-2\pi ikn/12}.
\]
\[
\widehat{c_{12}}(k)=\sum_{a\in U_{12}}\sum_{n=0}^{11}\zeta^{(a-k)n}
=12\,\mathbf1_{k\in U_{12}}.
\]
The inner geometric sum is twelve when \(a=k\) modulo twelve and zero otherwise. The selector and the Fourier support are now the same exact data.