منابع مشابه
Linear Analysis of Quadrature Domains. IV
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 understanding of the linear data associated in previous works to a quadrature domain. Version: September 10, 2003. 1. The exponential transform Let Ω be a bounded open subset of the co...
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Formulas like (0.1) and (0.2) are called quadrature identities, and the corresponding domains of integration are called (classical) quadrature domains. Various classes of quadrature domains have been known for quite some time, see e.g. Neumann’s papers [39, 40] from the beginning of the last century, but the systematic study began only with the work of Davis [13], and Aharonov and Shapiro [2]. ...
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It is well known that, in the plane, the boundary of any quadrature domain (in the classical sense) coincides with the zero set of a polynomial. We show, by explicitly constructing some four-dimensional examples, that this is not always the case. This confirms, in dimension 4, a conjecture of the second author. Our method is based on the Schwarz potential and involves elliptic integrals of the ...
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This is a selection of facts, old and new, about quadrature domains. The text, written in the form of a survey, is addressed to non-experts and covers a variety of phenomena related to quadrature domains. Such as: the difference between quadrature domains for subharmonic, harmonic and respectively complex analytic functions, geometric properties of the boundary, instability in the reverse Hele-...
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First we discuss briefly our former characterization theorem for positive interpolation quadrature formulas (abbreviated qf), provide an equivalent characterization in terms of Jacobi matrices, and give links and applications to other qf, in particular to Gauss-Kronrod quadratures and recent rediscoveries. Then for any polynomial tn which generates a positive qf, a weight function (depending on...
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ژورنال
عنوان ژورنال: Journal of Mathematical Analysis and Applications
سال: 1999
ISSN: 0022-247X
DOI: 10.1006/jmaa.1999.6556