Please use this identifier to cite or link to this item: http://hdl.handle.net/10016/5969

 Google™ Scholar. Others By: Bueno, M. Isabel - Marcellán, Francisco
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 Title: Darboux transformation and perturbation of linear functionals Author(s): Bueno, M. IsabelMarcellán, Francisco Publisher: Elsevier Issued date: Jun-2004 Citation: Linear Algebra and its Applications, 2004, vol. 384, p. 215-242 URI: http://hdl.handle.net/10016/5969 ISSN: 0024-3795 DOI: 10.1016/j.laa.2004.02.004 Description: 28 pages, no figures.-- MSC2000 codes: 42C05; 15A23.MR#: MR2055354 (2005b:15027)Zbl#: Zbl 1055.42016 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. Sponsor: 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. Review: PeerReviewed Publisher version: http://dx.doi.org/10.1016/j.laa.2004.02.004 Keywords: LU factorizationMonic Jacobi matrixOrthogonal polynomialsDarboux transformation Rights: © Elsevier Appears in Collections: DM - GAMA - Artículos de Revistas