# On 2D Inverse Problems/Fourier coordinates

Let ${\displaystyle \omega }$ be a not unit N'th root of unity, i.e. ${\displaystyle \omega ^{N}=1,\omega \neq 1}$.

${\displaystyle F_{\omega }={\frac {1}{\sqrt {N}}}{\begin{bmatrix}1&1&1&\ldots &1\\1&\omega &\omega ^{2}&\ldots &\omega ^{(N-1)}\\1&\omega ^{2}&\vdots &\ldots &\omega ^{2(N-1)}\\\vdots &\vdots &\vdots &\ddots &\vdots \\1&\omega ^{(N-1)}&\omega ^{2(N-1)}&\ldots &\omega ^{(N-1)^{2}}\\\end{bmatrix}}}$
For example,

${\displaystyle F_{e^{2\pi i/5}}={\frac {1}{\sqrt {5}}}{\begin{bmatrix}1&1&1&1&1\\1&\omega &\omega ^{2}&\omega ^{3}&\omega ^{4}\\1&\omega ^{2}&\omega ^{4}&\omega ^{6}&\omega ^{8}\\1&\omega ^{3}&\omega ^{6}&\omega ^{9}&\omega ^{12}\\1&\omega ^{4}&\omega ^{8}&\omega ^{12}&\omega ^{16}\\\end{bmatrix}}={\frac {1}{\sqrt {5}}}{\begin{bmatrix}1&1&1&1&1\\1&\omega &\omega ^{2}&\omega ^{3}&\omega ^{4}\\1&\omega ^{2}&\omega ^{4}&\omega &\omega ^{3}\\1&\omega ^{3}&\omega &\omega ^{4}&\omega ^{2}\\1&\omega ^{4}&\omega ^{3}&\omega ^{2}&\omega \\\end{bmatrix}}.}$
${\displaystyle F_{N}^{2}=Id.}$