User:DVD206/Printed Version/Fourier transform

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For a natural number N, let be the N'th root of unity:

.

We consider the following symmetric Vandermonde matrix, defining the discrete Fourier transform of a vector:

For example,

Exercise (*). Prove that the square of the Fourier transform is the flip permutation matrix:

and the forth power of the Fourier transform is the identity:

Exercise (**). Prove the following discrete Poisson summation formula, see [30]: Let k,l and N be positive integers, such that N=kl. For a natural number k let 1k be a vector in Rk w/every kth coordinate equal to 1, and 0 otherwise. Then

For a function f defined on a circle w/integrable square

the Fourier transform is function/vector on integers defined by the formula:

The function f can be recovered from its Fourier transform by the inversion formula:

Exercise (*). If a planar network or a domain is rotation invariant then its Dirichlet-to-Neumann operator can be diagonalized by Fourier transform in the discrete and continuous settings. (Hint) The harmonic functions commute w/rotation.