Please use this identifier to cite or link to this item: http://hdl.handle.net/10016/5976

 Google™ Scholar. Others By: Marcellán, Francisco - Szafraniec, Franciszek H.
Files in This Item:
 Title: Integral representations on equipotential and harmonic sets Author(s): Marcellán, FranciscoSzafraniec, Franciszek H. Publisher: The Belgian Mathematic Society Issued date: 2004 Citation: Bulletin of the Belgian Mathematical Society - Simon Stevin, 2004, vol. 11, n. 3, p. 457-468 URI: http://hdl.handle.net/10016/5976 ISSN: 1370-1444 Description: 12 pages, no figures.-- MSC1991 codes: Primary 46E35, 46E39, 46E20; Secondary 43A35, 44A60.MR#: MR2098419 (2005h:30066)Zbl#: Zbl 1082.46026 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. Sponsor: 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. Review: PeerReviewed Publisher version: http://projecteuclid.org/euclid.bbms/1093351384 Keywords: Inner product on the space of polynomialsMoment problemsSobolev inner productEquipotential and harmonic setsRecurrence relationMatrix integration Rights: © The Belgian Mathematic Society Appears in Collections: DM - GAMA - Artículos de Revistas