Orthogonal Polynomials with Respect to Modified Jacobi Weight and Corresponding Quadrature Rules of Gaussian Type
Abstract
In this paper we consider polynomials orthogonal with respect to the linear functional L : P -> C, defined on the space of all algebraic polynomials P by L[p] = integral(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 l = vertical bar beta - alpha vertical bar is a positive integer, and zeta is an element of R\{0}. We prove the existence of such orthogonal polynomials for some pairs of alpha and zeta and for all nonnegative integers l. 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.
Keywords:
recurrence relation / Orthogonal polynomials / modified Jacobi weight function / Gaussian quadrature ruleSource:
Numerical Mathematics, 2011, 4, 4, 478-488Publisher:
- Global Science Press
Funding / projects:
- Approximation of integral and differential operators and applications Апроксимација интегралних и диференцијалних оператора и примене Aproksimacija integralnih i diferencijalnih operatora i primene (RS-MESTD-Basic Research (BR or ON)-174015)
- Development of new information and communication technologies, based on advanced mathematical methods, with applications in medicine, telecommunications, power systems, protection of national heritage and education (RS-MESTD-Integrated and Interdisciplinary Research (IIR or III)-44006)
DOI: 10.4208/nmtma.2011.m103
ISSN: 1004-8979
WoS: 000298766300003
Scopus: 2-s2.0-84856491451
Collections
Institution/Community
Mašinski fakultetTY - JOUR AU - Stanić, Marija P. AU - Cvetković, Aleksandar PY - 2011 UR - https://machinery.mas.bg.ac.rs/handle/123456789/1141 AB - In this paper we consider polynomials orthogonal with respect to the linear functional L : P -> C, defined on the space of all algebraic polynomials P by L[p] = integral(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 l = vertical bar beta - alpha vertical bar is a positive integer, and zeta is an element of R\{0}. We prove the existence of such orthogonal polynomials for some pairs of alpha and zeta and for all nonnegative integers l. 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. PB - Global Science Press T2 - Numerical Mathematics T1 - Orthogonal Polynomials with Respect to Modified Jacobi Weight and Corresponding Quadrature Rules of Gaussian Type EP - 488 IS - 4 SP - 478 VL - 4 DO - 10.4208/nmtma.2011.m103 ER -
@article{ author = "Stanić, Marija P. and Cvetković, Aleksandar", year = "2011", abstract = "In this paper we consider polynomials orthogonal with respect to the linear functional L : P -> C, defined on the space of all algebraic polynomials P by L[p] = integral(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 l = vertical bar beta - alpha vertical bar is a positive integer, and zeta is an element of R\{0}. We prove the existence of such orthogonal polynomials for some pairs of alpha and zeta and for all nonnegative integers l. 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.", publisher = "Global Science Press", journal = "Numerical Mathematics", title = "Orthogonal Polynomials with Respect to Modified Jacobi Weight and Corresponding Quadrature Rules of Gaussian Type", pages = "488-478", number = "4", volume = "4", doi = "10.4208/nmtma.2011.m103" }
Stanić, M. P.,& Cvetković, A.. (2011). Orthogonal Polynomials with Respect to Modified Jacobi Weight and Corresponding Quadrature Rules of Gaussian Type. in Numerical Mathematics Global Science Press., 4(4), 478-488. https://doi.org/10.4208/nmtma.2011.m103
Stanić MP, Cvetković A. Orthogonal Polynomials with Respect to Modified Jacobi Weight and Corresponding Quadrature Rules of Gaussian Type. in Numerical Mathematics. 2011;4(4):478-488. doi:10.4208/nmtma.2011.m103 .
Stanić, Marija P., Cvetković, Aleksandar, "Orthogonal Polynomials with Respect to Modified Jacobi Weight and Corresponding Quadrature Rules of Gaussian Type" in Numerical Mathematics, 4, no. 4 (2011):478-488, https://doi.org/10.4208/nmtma.2011.m103 . .