The continuous analog of the matrix representation of a Dirichlet-to-Neumann operator for a domain is its kernel. It is a distribution defined on the Cartesian product of the boundary of the domain w/itself, such that if
where and parametrise the boundary.
For the case of a half-plane w/unit uniform conductivity one can calculate the kernel explicitly. The kernel is a convolution, because the domain in consideration is shift invariant,
where k is a distribution on a line. Therefore, the calculation reduces to solving the Dirichlet problem for a -function at the origin and taking normal derivative at the boundary line.
Exercise (**). Complete the calculation of the kernel K for the half-plane to show that:
off the diagonal.
Exercise (*). Prove that for rotation invariant domain (disc w/ conductivity depending only on radius) the kernel of Dirichlet-to-Neumann map is a convolution.
The Hilbert transform gives a correspondence between boundary values of harmonic function and its harmonic conjugate.
is an analytic function in the domain.
For the case of the complex upper 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, one considers the corresponding graph together w/its dual.