Integral representations on equipotential and harmonic sets

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Show simple item record Marcellán Español, Francisco José Szafraniec, Franciszek H. 2009-12-07T12:49:34Z 2009-12-07T12:49:34Z 2004
dc.identifier.bibliographicCitation Bulletin of the Belgian Mathematical Society - Simon Stevin, 2004, vol. 11, n. 3, p. 457-468
dc.identifier.issn 1370-1444
dc.description 12 pages, no figures.-- MSC1991 codes: Primary 46E35, 46E39, 46E20; Secondary 43A35, 44A60.
dc.description MR#: MR2098419 (2005h:30066)
dc.description Zbl#: Zbl 1082.46026
dc.description.abstract The sets we are going to consider here are of the form ${z\in\mathbb C \mid (z) 1}$ (equipotential) and ${z\in\mathbb C \mid IM A(z)=0}$ (harmonic) with $A$ being a polynomial with complex coefficients. There are two themes which we want to focus on and which come out from invariance property of inner products on $\mathbb C[Z]$ related to the aforesaid sets. First, we formalize the construction of integral representation of the inner products in question with respect to matrix measure. Then we show that these inner products when represented in a Sobolev way are precisely those with discrete measures in the higher order terms of the representation. In this way we fill up the case already considered in [3] by extending it from the real line to harmonic sets on the complex plane as well as we describe completely what happens in this matter on equipotential sets. As a kind of smooth introduction to the above we are giving an account of standard integral representations on the complex plane in general and of those supported by these two kinds of real algebraic sets.
dc.description.sponsorship The work of the first author was partially supported by Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain under the grant BFM 2000-0206-C04-01 and by INTAS under the grant INTAS 2000-272. The final stage of the work was done during the second author’s visit to Universidad Carlos III de Madrid under the bilateral cooperation programme in culture and education between Spain and Poland, April 2002.
dc.format.mimetype application/pdf
dc.language.iso eng
dc.publisher The Belgian Mathematic Society
dc.rights © The Belgian Mathematic Society
dc.subject.other Inner product on the space of polynomials
dc.subject.other Moment problems
dc.subject.other Sobolev inner product
dc.subject.other Equipotential and harmonic sets
dc.subject.other Recurrence relation
dc.subject.other Matrix integration
dc.title Integral representations on equipotential and harmonic sets
dc.type article PeerReviewed
dc.description.status Publicado
dc.subject.eciencia Matemáticas
dc.rights.accessRights openAccess
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