Bergman Orthogonal Polynomials on Archipelaga: Construction, Asymptotics, Short Recurrences and Shape Recovery
Let G := [Nj
=1Gj be the union of N mutually exterior, bounded domains in the complex plane and let
n=0 denote the sequence of Bergman polynomials of G. This is dened as the sequence
pn(z) = lnzn+ ; ln > 0; n = 0;1;2; : : : ;
of polynomials that are orthonormal with respect to the inner product h f ;gi :=
G f (z)g(z)dA(z); where dA
stands for the area measure. (In the case when N > 1, we call G an archipelago.)
The purpose of the talk is to present some recent developments regarding the construction, the theory
and the applications of Bergman polynomials. These developments include:
(i) A stable Arnoldi Gram-Schmidt process for constructing orthonormal polynomials.
(ii) Ratio asymptotics and nite-term recurrence relations for Bergman polynomials.
(iii) An reconstruction algorithm for recovering the shape of G in the single component case N = 1 from
a nite section of its complex moment matrix,
mi j :=
ziz jdA(z); 0 i; j n;
using ratio asymptotics. The importance of the algorithm in (iii) is underlined by the fact that suitable
tomographic data, for example, parallel ray measurements in 2D geometric tomography, can be transformed
into a nite section of the moment matrix.