Extremal Solutions of the Two-DimensionalL-Problem of Moments, II

Putinar, Mihai

doi:10.1006/jath.1997.3117

Cited by 24 publications

(7 citation statements)

References 15 publications

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“…The difficulties are of a different nature, although the basic moment theory is essentially unchanged. A parallel correspondence between Krein's interpretation of the L problem on the line, via the perturbation theory of self-adjoint operators, was recently proposed in [24]; this time, the L problem was interpreted as the inverse problem for the principal function of a pair of self-adjoint operators with trace-class commutator. This dictionary provides a simple solution to the L problem in the plane, and gives the technical tool (a formal exponential transform of the moment sequence) in reconstructing a domain from its moments, see also [12,13,23].…”

Section: Introductionmentioning

confidence: 99%

Reconstructing planar domains from their moments

et al. 2000

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In many areas of science and engineering it is of interest to find the shape of an object or region from indirect measurements which can actually be distilled into moments of the underlying shapes we seek to reconstruct. In this paper, we describe a theoretical framework for the reconstruction of a class of planar semi-analytic domains from their moments. A part of this class, known as quadrature domains, can approximate, arbitrarily closely, any bounded domain in the complex plane, and is therefore of great practical importance. We provide an exact reconstruction algorithm of quadrature domains. Some numerical demonstrations of the proposed algorithms will be presented. In addition, relations of the present theory to computer-assisted tomography and a geophysical inverse problem will be briefly discussed.

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Section: Introductionmentioning

confidence: 99%

Reconstructing planar domains from their moments

et al. 2000

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“…For an historical account on this problem as well as other developments, the interested reader is referred to e.g. Krein [21], Krein and Nuldelman [22], Karlin and Studden [20] and Putinar [37].…”

Section: Introductionmentioning

confidence: 99%

Approximate Volume and Integration for Basic Semialgebraic Sets

Henrion¹,

Lasserre²,

Savorgnan³

2009

SIAM Rev.

235

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Given a basic compact semi-algebraic set K ⊂ R n , we introduce a methodology that generates a sequence converging to the volume of K. This sequence is obtained from optimal values of a hierarchy of either semidefinite or linear programs. Not only the volume but also every finite vector of moments of the probability measure that is uniformly distributed on K can be approximated as closely as desired, and so permits to approximate the integral on K of any given polynomial; extension to integration against some weight functions is also provided. Finally, some numerical issues associated with the algorithms involved are briefly discussed.

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“…Subnormal operators with finite rank self-commutators have been extensively studied [2,21,24,[29][30][31]33,34]. Particular attention has been paid to hyponormal operators with rank-one or rank-two self-commutators [19,23,[25][26][27]29,32,35]. In particular, B. Morrel [23] showed that a pure subnormal operator with rank-one self-commutator (pure means having no normal summand) is unitarily equivalent to a linear function of the unilateral shift.…”

Section: Introductionmentioning

confidence: 98%

Hyponormal operators with rank-two self-commutators

Lee

2009

Journal of Mathematical Analysis and Applications

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In this paper it is shown that if T ∈ L(H) satisfies (i) T is a pure hyponormal operator;then T is either a subnormal operator or the Putinar's matricial model of rank two. More precisely, if T | ker[T * ,T ] has a rank-one self-commutator then T is subnormal and if instead T | ker[T * ,T ] has a rank-two self-commutator then T is either a subnormal operator or the kth minimal partially normal extension, T k (k) , of a (k + 1)-hyponormal operator T k which has a rank-two self-commutator for any k ∈ Z + . Hence, in particular, every weakly subnormal (or 2-hyponormal) operator with a rank-two self-commutator is either a subnormal operator or a finite rank perturbation of a k-hyponormal operator for any k ∈ Z + .

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Extremal Solutions of the Two-DimensionalL-Problem of Moments, II

Cited by 24 publications

References 15 publications

Reconstructing planar domains from their moments

Reconstructing planar domains from their moments

Approximate Volume and Integration for Basic Semialgebraic Sets

Hyponormal operators with rank-two self-commutators

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