Applications of the Bergman projection to quadrature domains and the Khavinson-Shapiro conjecture
This thesis employs complex analysis via the Bergman projection and kernel to examine two themes: quadrature domains and the Khavinson-Shapiro conjecture. Quadrature domains for analytic functions of several variables are introduced, with some emphasis given to the case of polynomial inclusion in the Bergman span, and to differences from the case of the plane. Then it is shown that every smooth bounded convex domain is biholomorphic to a quadrature domain. In the plane it is shown that continuous deformation can sometimes be accomplished through quadrature domains. Then, the possibility of a relationship between the Bergman projection, polynomials, and the Khavinson-Shapiro conjecture is studied in one and several complex variables. The polyharmonic Bergman projections in real space and a weighted Szego projection in the plane are shown to map polynomials to polynomials on ellipsoidal domains.
Bell, Purdue University.
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