# Darboux transformation and perturbation of linear functionals

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 dc.contributor.author Bueno Cachadiña, María Isabel dc.contributor.author Marcellán Español, Francisco José dc.date.accessioned 2009-12-07T09:37:04Z dc.date.available 2009-12-07T09:37:04Z dc.date.issued 2004-06 dc.identifier.bibliographicCitation Linear Algebra and its Applications, 2004, vol. 384, p. 215-242 dc.identifier.issn 0024-3795 dc.identifier.uri http://hdl.handle.net/10016/5969 dc.description 28 pages, no figures.-- MSC2000 codes: 42C05; 15A23. dc.description MR#: MR2055354 (2005b:15027) dc.description Zbl#: Zbl 1055.42016 dc.description.abstract Let L be a quasi-definite linear functional defined on the linear space of polynomials with real coefficients. In the literature, three canonical transformations of this functional are studied: $\bold{xL}$, $\bold L+\bold C\delta (\bold x)$ and $\frac {1}{\bold x}\bold L +\bold C\delta(\bold x)$ where $\delta(x)$ denotes the linear functional $(\delta(x))(x )=\delta_{k,0}$, and $\delta_{k,0}$ is the Kronecker symbol. Let us consider the sequence of monic polynomials orthogonal with respect to L. This sequence satisfies a three-term recurrence relation whose coefficients are the entries of the so-called monic Jacobi matrix. In this paper we show how to find the monic Jacobi matrix associated with the three canonical perturbations in terms of the monic Jacobi matrix associated with L. The main tools are Darboux transformations. In the case that the LU factorization of the monic Jacobi matrix associated with xL does not exist and Darboux transformation does not work, we show how to obtain the monic Jacobi matrix associated with $\bold x \bold L$ as a limit case. We also study perturbations of the functional L that are obtained by combining the canonical cases. Finally, we present explicit algebraic relations between the polynomials orthogonal with respect to L and orthogonal with respect to the perturbed functionals. dc.description.sponsorship The work of the authors has been partially supported by Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain under grant BFM 2003-06335-C03-02 and NATO collaborative grant PST.CLG.979738. dc.format.mimetype application/pdf dc.language.iso eng dc.publisher Elsevier dc.rights © Elsevier dc.subject.other LU factorization dc.subject.other Monic Jacobi matrix dc.subject.other Orthogonal polynomials dc.subject.other Darboux transformation dc.title Darboux transformation and perturbation of linear functionals dc.type article dc.type.review PeerReviewed dc.description.status Publicado dc.relation.publisherversion http://dx.doi.org/10.1016/j.laa.2004.02.004 dc.subject.eciencia Matemáticas dc.identifier.doi 10.1016/j.laa.2004.02.004 dc.rights.accessRights openAccess
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