On 2D Inverse Problems/An infinite example

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The following construction provides an example of an infinite network (featured on the cover of the book), which Dirichlet-to-Neumann operator satisfies the equation in the title of this chapter. 

The matrix equation reflects the self-duality and self-symmetry of the network. 
Exercise (**). Prove that the Dirichlet-to-Neumann operator of the network on the picture w/the natural boundary satisfies the equation.
The self-dual self-symmetric infinite graph w/its dual

(Hint:) Use the fact that the operator/matrix is the fixed point of the Schur complement]]: where is the circulant matrix, such that