Darboux transformation and perturbation of linear functionals

28 pages, no figures.-- MSC2000 codes: 42C05; 15A23.MR#: MR2055354 (2005b:15027)Zbl#: Zbl 1055.42016Let 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.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.Publicad