Darboux transformation and perturbation of linear functionals

e-Archivo Repository

Show simple item record

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
 Find Full text

Files in this item

*Click on file's image for preview. (Embargoed files's preview is not supported)

This item appears in the following Collection(s)

Show simple item record