View source for On 2D Inverse Problems/Kernel of Dirichlet-to-Neumann map

 The continuous analog of the matrix representation of a Dirichlet-to-Neumann operator for a domain <math>\Omega</math> is its '''kernel'''. It is a distribution defined on the '''Cartesian product''' of the boundary of the domain w/itself, such that if  
<math>
\Lambda f = g,
</math>
then
<math>
g(\phi) = \int_{\partial\Omega}K(\phi,\theta)f(\theta)d\theta,
</math> where <math>\phi</math> and <math>\theta</math> parametrize the boundary w/arclength measure. 

 For the case of the half-plane w/constant unit conductivity the kernel can be calculated explicitly. It's a convolution, because the domain in consideration is shift invariant:
<math>
K(\phi,\theta) = k(\phi-\theta),
</math> where ''k'' is a distribution on a line.
Therefore, the calculation reduces to solving the '''Dirichlet problem''' for a <math>\delta_0</math>-function at the origin and taking outward derivative at the boundary line.
'''
[[File:Harmonic measure (1).jpg|center|550px|Dirichlet problem for a half-plane]]
'''

'''Exercise (**)'''. Complete the calculation of the kernel ''K'' for the half-plane to show that <math>K(x,y) = \frac{-1}{\pi(x-y)^2}</math> 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 a harmonic function and its harmonic conjugate.
<math>
H:u|_{\partial\Omega}\rightarrow v|_{\partial\Omega},
</math>
where
<math>f(z) = u(z) + iv(z)</math>
is an analytic function in the domain.

For the case of the complex upper half-plane the Hilbert transform is given by the following formula:
<math>Hf(y) =\frac{1}{\pi} \ \text{p.v.} \int_{-\infty}^{\infty} \frac{f(x)}{y-x}dx.</math>

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

 To define discrete Hilbert transform for a planar network, the network together w/its dual is needed.

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