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

then

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.

where

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.