On 2D Inverse Problems/Fourier coordinates

Let $$\omega$$ be a not unit N'th root of unity, i.e. $$\omega^N = 1, \omega \ne 1$$. The discrete Fourier transform 's given by the symmetric Vandermonde matrix:


 * $$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,


 * $$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}. $$


 * Exercise (*). The square of the Fourier transform is the identity transform:
 * $$F_N^2 = Id.$$


 * Exercise (*). If an e-network is rotation invariant, then so 's the conductivity equation and the Dirichlet-to-Neumann map is diagonal in the Fourier coordinates (the column vectors of the matrix.