Growth estimates for orthogonal polynomials with
respect to area measure (Bergman polynomials) over the union of
finitely many Jordan regions with piecewise smooth boundary are
obtained by a careful investigation of the Green function of the
complement, and of Schwarz reflection in analytic arcs of the
boundary. As applications one derives a detailed picture of the
limiting zero distribution of Bergman's orthogonal polynomials,
and also a robust reconstruction algorithm of the
original open set, starting from incomplete data (such as obtained
by geometric tomography). A good part of the lecture will be devoted
to a non-technical discussion of the main objects and principal
techniques used in the above study, from a historical perspective.
Several numerical experiments and illustrations will be provided,
as supports for the theoretical facts. Based on recent joint work
with B. Gustafsson, E. Saff and N. Stylianopoulos.