User:DVD206/Printed Version/Fourier transform

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


 * $$\omega^N = e^{\frac{2\pi i}{N}}.

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


 * $$\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}. $$

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


 * $$\mathbf{F}_N^2 = P,$$

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


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

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



1_k\mathbf{F}_N = \frac{k}{\sqrt{N}}1_l. $$

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


 * $$\int_0^{2\pi}|f(\theta)|^2d\theta < \infty $$

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


 * $$\mathbf{F}(k) = \frac{1}{2 \pi}\int_0^{2\pi}f(\theta)e^{-i k\theta}d\theta. $$

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


 * $$f(\theta) = \sum_{k = -\infty}^{\infty}\mathbf{F}(k)e^{i k\theta} $$

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.