Linear analysis of quadrature domains. II
A bounded domain \(\Omega\) in the complex plane is called a quadrature domain if a distribution \(u\) with finite support in \(\Omega\) exists so that \(\int_\Omega fdA=u(f)\) for all \(f\) holomorphic and integrable in \(\Omega\), where \(A\) denotes the two-dimensional Lebesgue measure. The paper is a continuation of a work of \textit{M. Putinar} [Ark. Mat. 33, No. 2, 357-376 (1995; Zbl 0892.47025)] and is devoted to some constructive aspects of the relation between quadrature domains and their linear data. The first sections are concerned with formulas for the moments \(\int_\Omega z^m\overline z^n dA(z)\) of the domain, where the defining polynomial of \(\Omega\) is assumed to be given. The further sections of the paper deal with some properties of the resolvent of the linear data of a quadrature domain. Finally, explicit computations of order-two quadrature domains illustrate the main results.
scientific article; zbMATH DE number 1558324
scientific article