Computational complexity. On the geometry of polynomials and a theory of cost. I
This work deals with a two-dimensional inverse problem in the field of tomography. The geometry of an unknown inclusion has to be reconstructed from boundary measurements. In this paper, we extend previous results of R. Kress and his coauthors: the leading idea is to use the conformal mapping function as unknown. We establish an integrodifferential equation that the trace of the Riemann map solves. We write it as a fixed point equation and give conditions for contraction. We conclude with a series...
The conformal mapping ω(z) of a domain D onto the unit disc must satisfy the condition |ω(t)| = 1 on ∂D, the boundary of D. The last condition can be considered as a Dirichlet problem for the domain D. In the present paper this problem is reduced to a system of functional equations when ∂D is a circular polygon with zero angles. The mapping is given in terms of a Poincaré series.