Orthogonal Polynomials with Respect to Modified Jacobi Weight and Corresponding Quadrature Rules of Gaussian Type
Year: 2011
Author: Aleksandar S. Cvetković, Marija P. Stanić, Aleksandar S. Cvetković
Numerical Mathematics: Theory, Methods and Applications, Vol. 4 (2011), Iss. 4 : pp. 478–488
Abstract
In this paper we consider polynomials orthogonal with respect to the linear functional $\mathcal{L}:\mathcal{P}\to \mathbb{C}$, defined on the space of all algebraic polynomials $\mathcal{P}$ by$$\mathcal{L}[p] =\int_{-1}^1 p(x) (1-x)^{\alpha-1/2} (1+x)^{\beta-1/2}\exp(i\zeta x)dx,$$ where $\alpha,\beta >-1/2$ are real numbers such that $\ell=|\beta-\alpha|$ is a positive integer, and $\zeta\in\mathbb{R}\backslash\{0\}$. We prove the existence of such orthogonal polynomials for some pairs of $\alpha$ and $\zeta$ and for all nonnegative integers $\ell$. For such orthogonal polynomials we derive three-term recurrence relations and also some differential-difference relations. For such orthogonal polynomials the corresponding quadrature rules of Gaussian type are considered. Also, some numerical examples are included.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/nmtma.2011.m1039
Numerical Mathematics: Theory, Methods and Applications, Vol. 4 (2011), Iss. 4 : pp. 478–488
Published online: 2011-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 11
Keywords: Orthogonal polynomials modified Jacobi weight function recurrence relation Gaussian quadrature rule.