User:Daviddaved/Hilbert transfrom

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The Hilbert transform gives a correspondence between boundary values of harmonic function and its harmonic conjugate.


H:u|_{\partial\Omega}\rightarrow v|_{\partial\Omega},

where

f(z) = u(z) + iv(z)

is an analytic function in the domain.

Exercise (*). Prove that for the case of the complex half-plane C+ the Hilbert transform is given by the following formula:

H_{C^+}f(y) =\frac{1}{\pi} \ \text{p.v.} \int_{-\infty}^{\infty} \frac{f(x)}{y-x}dx.

Exercise (*). Differentiate under integral sign the formula above to obtain the kernel representation for the Dirichlet-to-Neumann operator for the uniform half plane.

To define discrete Hilbert transform for a planar network, we need to consider it together w/its dual.