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Linear analysis of quadrature domains. II. (English) Zbl 0968.30016
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 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.

##### MSC:
 3e+06 Moment problems and interpolation problems in the complex plane