# Integral representations on equipotential and harmonic sets

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 dc.contributor.author Marcellán Español, Francisco José dc.contributor.author Szafraniec, Franciszek H. dc.date.accessioned 2009-12-07T12:49:34Z dc.date.available 2009-12-07T12:49:34Z dc.date.issued 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.identifier.uri http://hdl.handle.net/10016/5976 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 dc.type.review PeerReviewed dc.description.status Publicado dc.relation.publisherversion http://projecteuclid.org/euclid.bbms/1093351384 dc.subject.eciencia Matemáticas dc.rights.accessRights openAccess
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