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