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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:
30E05 Moment problems and interpolation problems in the complex plane
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[1] [AS] D. Aharonov and H. S. Shapiro,Domains on which analytic functions satisfy quadrature identities, Journal d’Analyse Mathématique30 (1976), 39–73. · Zbl 0337.30029 · doi:10.1007/BF02786704
[2] [BGR] J. A. Ball, I. Gohberg and L. Rodman,Interpolation of Rational Matrix Functions, Birkhäuser Verlag, Basel, 1990.
[3] [D] Ph. J. Davis,The Schwarz function and its applications, Carus Mathematical Monographs Vol, 17, Mathematical Association of America, 1974. · Zbl 0293.30001
[4] [DL] J. Dennis and J. Lawrence,A Catalog of Special Plane Curves, Dover, New York, 1972. · Zbl 0257.50002
[5] [G1] B. Gustafsson,Quadrature identities and the Schottky double, Acta Applicandae Mathematicae1 (1983), 209–240. · Zbl 0559.30039 · doi:10.1007/BF00046600
[6] [G2] B. Gustafsson,Singular and special points on quadrature domains from an algebraic point of view, Journal d’Analyse Mathématique51 (1988), 91–117. · Zbl 0656.30034 · doi:10.1007/BF02791120
[7] [GH] Ph. Griffiths and J. Harris,Principles of Algebraic Geometry, Wiley, New York, 1994. · Zbl 0836.14001
[8] [KS] D. Khavinson and H. S. Shapiro,The Vekua hull of a plane domain, Complex Variables14 (1990), 117–128. · Zbl 0705.35018
[9] [P1] M. Putinar,Linear analysis of quadrature domains, Arkiv för Matematik33 (1995), 357–376. · Zbl 0892.47025 · doi:10.1007/BF02559714
[10] [P2] M. Putinar,Extremal solutions of the two dimensional L-problems of moments, Journal of Functional Analysis136 (1996), 331–364. · Zbl 0917.47014 · doi:10.1006/jfan.1996.0033
[11] [Sa] M. Sakai,Regularity of boundaries of quadrature domains in two dimensions, SIAM Journal on Mathematical Analysis24 (1993), 341–364. · Zbl 0771.30041 · doi:10.1137/0524023
[12] [Sh] H. S. Shapiro,The Schwarz Function and its Generalization to Higher Dimensions, University of Arkansas Lecture Notes in Mathematics, Vol. 9, Wiley, New York, 1992. · Zbl 0784.30036
[13] [ST] J. A. Shohat and J. D. Tamarkin,The Problem of Moments, American Mathematical Society, Providence, RI, 1943. · Zbl 0063.06973
[14] [V] I. N. Vekua,New Methods for Solving Elliptic Equations, North-Holland Series in Applied Mathematics and Mechanics No. 1, North-Holland, Amsterdam, 1967 (translated from Russian). · Zbl 0146.34301
[15] [Xu] Y. Xu,On orthogonal polynomials in several variables, Fields Institute Communications14 (1997), 247–270. · Zbl 0873.42016
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