# Differential properties for Sobolev orthogonality on the unit circle

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 dc.contributor.author Berriochoa, Elías dc.contributor.author Cachafeiro, Alicia dc.contributor.author Marcellán Español, Francisco José dc.date.accessioned 2009-12-16T12:59:27Z dc.date.available 2009-12-16T12:59:27Z dc.date.issued 2001-08-01 dc.identifier.bibliographicCitation Journal of Computational and Applied Mathematics, 2001, vol. 133, n. 1-2, p. 231-239 dc.identifier.issn 0377-0427 dc.identifier.uri http://hdl.handle.net/10016/6125 dc.description 9 pages, no figures.-- MSC2000 code: 42C05.-- Issue title: Proceedings of the Fifth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Patras, 1999). dc.description MR#: MR1858282 (2002m:42022) dc.description Zbl#: Zbl 0990.42006 dc.description.abstract The aim of this paper is to study differential properties of the sequence of monic orthogonal polynomials with respect to the following Sobolev inner product: $$\langle f, g\rangle_s= \int^{2\pi}_0 f(e^{i\theta}) \overline{g(e^{i\theta})} d\mu(\theta)+{1\over \lambda} \int^{2\pi}_0 f'(e^{i\theta}) \overline{g'(e^{i\theta})} {d\theta\over 2\pi},$$ where $\mu$ is a finite positive Borel measure on $[0, 2\pi]$ verifying the following conditions: the Carathéodory function associated with $\mu$ has an analytic extension outside the unit disk and the induced norm is equivalent to the Lebesgue norm in the space $L_2$. Here $d\theta/2\pi$ is the normalized Lebesgue measure and $\lambda$ is a positive real number. The nonhomogeneous second-order differential equations satisfied by the sequence of monic Sobolev orthogonal polynomials are obtained. Moreover, as an application, a sample of the Dirichlet boundary value problem is solved. dc.description.sponsorship The research was supported by DGES under grants number PB96-0344 and PB96-0120 C03-01. dc.format.mimetype application/pdf dc.language.iso eng dc.publisher Elsevier dc.rights © Elsevier dc.subject.other Orthogonal polynomials dc.subject.other Sobolev inner products dc.subject.other Differential operators dc.title Differential properties for Sobolev orthogonality on the unit circle dc.type article dc.type.review PeerReviewed dc.description.status Publicado dc.relation.publisherversion http://dx.doi.org/10.1016/S0377-0427(00)00645-2 dc.subject.eciencia Matemáticas dc.identifier.doi 10.1016/S0377-0427(00)00645-2 dc.rights.accessRights openAccess
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