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Linear Analysis of Quadrature Domains. IV Bjorn Gustafsson and Mihai Putinar

Summary: Linear Analysis of Quadrature Domains. IV
Bjšorn Gustafsson and Mihai Putinar
Paper dedicated to Harold S. Shapiro on the occasion of his seventy-fifth birthday
Abstract. The positive definiteness of the exponential transform of a planar
domain is proved by elementary means. This direct approach avoids the heavy
machinery of the theory of hyponormal operators and leads to a better un-
derstanding of the linear data associated in previous works to a quadrature
Version: September 10, 2003.
1. The exponential transform
Let be a bounded open subset of the complex plane and let dA stand for the
Lebesgue planar measure. The exponential transform of the set is the function:
(1.1) E(z, w) = exp[-

( - z)( - w)
The integral is convergent for all values of z, w C avoiding the diagonal
= {(z, w); z = w }.


Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara


Collections: Mathematics