User:Daviddaved/Fourier coordinates

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

.

We consider the following symmetric Vandermonde matrix:

For example,

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

The forth power of the Fourier transform is the identity:

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

,

where P is a cyclic permutation matrix.

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