Article
On a modification of the Jacobi linear functional asymptotic properties and zeros of the corresponding orthogonal polynomials
Author/s | Arvesú Carballo, Jorge
Marcellán Español, Francisco Álvarez Nodarse, Renato |
Department | Universidad de Sevilla. Departamento de Análisis Matemático |
Publication Date | 2002-04 |
Deposit Date | 2016-07-21 |
Published in |
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Abstract | The paper deals with orthogonal polynomials in the case where the orthogonality condition is related to semiclassical functionals. The polynomials that we discuss are a generalization of Jacobi polynomials and Jacobi-type ... The paper deals with orthogonal polynomials in the case where the orthogonality condition is related to semiclassical functionals. The polynomials that we discuss are a generalization of Jacobi polynomials and Jacobi-type polynomials. More precisely, we study some algebraic properties as well as the asymptotic behaviour of polynomials orthogonal with respect to the linear functional U U = Jα,β + A1δ(x − 1) + B1δ(x + 1) − A2δ (x − 1) − B2δ (x + 1), where Jα,β is the Jacobi linear functional, i.e. Jα,β , p = 1 −1 p(x)(1 − x)α(1 + x)β dx, α, β > −1, p ∈ P, and P is the linear space of polynomials with complex coefficients. The asymptotic properties are analyzed in (−1, 1) (inner asymptotics) and C \ [−1, 1] (outer asymptotics) with respect to the behaviour of Jacobi polynomials. In a second step, we use the above results in order to obtain the location of zeros of such orthogonal polynomials. Notice that the linear functional U is a generalization of one studied by T. H. Koornwinder when A2 = B2 = 0. From the point of view of rational approximation, the corresponding Markov function is a perturbation of the Jacobi–Markov function by a rational function with two double poles at ±1. The denominators of the [n−1/n] Padé approximants are our orthogonal polynomials. |
Funding agencies | Dirección General de Enseñanza Superior. España |
Project ID. | BFM2000-0029
BFM2000-0206-C04-01 BFM2000-0206-C04-02 |
Citation | Arvesú Carballo, J., Marcellán Español, F. y Álvarez Nodarse, R. (2002). On a modification of the Jacobi linear functional asymptotic properties and zeros of the corresponding orthogonal polynomials. Acta Applicandae Mathematica, 71 (2), 127-158. |
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