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 Title: Differential properties for Sobolev orthogonality on the unit circle Author(s): Berriochoa, ElíasCachafeiro, AliciaMarcellán, Francisco Publisher: Elsevier Issued date: 1-Aug-2001 Citation: Journal of Computational and Applied Mathematics, 2001, vol. 133, n. 1-2, p. 231-239 URI: http://hdl.handle.net/10016/6125 ISSN: 0377-0427 DOI: 10.1016/S0377-0427(00)00645-2 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).MR#: MR1858282 (2002m:42022)Zbl#: Zbl 0990.42006 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. Sponsor: The research was supported by DGES under grants number PB96-0344 and PB96-0120 C03-01. Review: PeerReviewed Publisher version: http://dx.doi.org/10.1016/S0377-0427(00)00645-2 Keywords: Orthogonal polynomialsSobolev inner productsDifferential operators Rights: © Elsevier Appears in Collections: DM - GAMA - Artículos de Revistas