User:DVD206/Fourier coordinates

For an integer N, let $\omega$ be the N'th root of unity, that is not equal to 1.

$\omega ^{N}=1,\omega \neq 1$.

We consider the following symmetric Vandermonde matrix:

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

For example,

$\mathbf {F_{5}} ={\frac {1}{\sqrt {5}}}{\begin{bmatrix}1&1&1&1&1\\1&\omega &\omega ^{2}&\omega ^{3}&\omega ^{4}\\1&\omega ^{2}&\omega ^{4}&\omega ^{6}&\omega ^{8}\\1&\omega ^{3}&\omega ^{6}&\omega ^{9}&\omega ^{12}\\1&\omega ^{4}&\omega ^{8}&\omega ^{12}&\omega ^{16}\\\end{bmatrix}}={\frac {1}{\sqrt {5}}}{\begin{bmatrix}1&1&1&1&1\\1&\omega &\omega ^{2}&\omega ^{3}&\omega ^{4}\\1&\omega ^{2}&\omega ^{4}&\omega &\omega ^{3}\\1&\omega ^{3}&\omega &\omega ^{4}&\omega ^{2}\\1&\omega ^{4}&\omega ^{3}&\omega ^{2}&\omega \\\end{bmatrix}}.$

The square of the Fourier transform is the flip permutation matrix:

$\mathbf {F} ^{2}=\mathbf {P} .$

The forth power of the Fourier transform is the identity:

$\mathbf {F} ^{4}=\mathbf {I} .$

Exercise (**). Proof that if N is a prime number than for any 0 < k < N

$\mathbf {F(\omega ^{k})} =\mathbf {P} \mathbf {F(\omega )} \mathbf {P} ^{T}$,

where P is a cyclic permutation matrix.

If a network is rotation invariant then its Dirichlet-to-Neumann operator is diagonal in Fourier coordinates.