 
Summary: Linear Analysis of Quadrature Domains. IV
Bjšorn Gustafsson and Mihai Putinar
Paper dedicated to Harold S. Shapiro on the occasion of his seventyfifth 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
domain.
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[
1
dA()
(  z)(  w)
].
The integral is convergent for all values of z, w C avoiding the diagonal
= {(z, w); z = w }.
