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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

On the convergence
of certain Gauss-type quadrature formulas
for unbounded intervals


Authors: A. Bultheel, C. Díaz-Mendoza, P. González-Vera and R. Orive
Journal: Math. Comp. 69 (2000), 721-747
MSC (1991): Primary 65D30; Secondary 41A21
Published electronically: February 24, 1999
MathSciNet review: 1651743
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider the convergence of Gauss-type quadrature formulas for the integral $\int _0^\infty f(x)\omega(x)\mathrm{d}x$, where $\omega$ is a weight function on the half line $[0,\infty)$. The $n$-point Gauss-type quadrature formulas are constructed such that they are exact in the set of Laurent polynomials $\Lambda _{-p,q-1}=\{\sum _{k=-p}^{q-1} a_k x^k$}, where $p=p(n)$ is a sequence of integers satisfying $0\le p(n)\le 2n$ and $q=q(n)=2n-p(n)$. It is proved that under certain Carleman-type conditions for the weight and when $p(n)$ or $q(n)$ goes to $\infty$, then convergence holds for all functions $f$ for which $f\omega$ is integrable on $[0,\infty)$. Some numerical experiments compare the convergence of these quadrature formulas with the convergence of the classical Gauss quadrature formulas for the half line.


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Additional Information

A. Bultheel
Affiliation: Department of Computer Science, K.U. Leuven, Celestijnenlaan 200A, B-3001 Heverlee, Belgium
Email: Adhemar.Bultheel@cs.kuleuven.ac.be

C. Díaz-Mendoza
Affiliation: Department Mathematical Analysis, Univ. La Laguna, Tenerife, Canary Islands, Spain
Email: cjdiaz@ull.es

P. González-Vera
Affiliation: Department Mathematical Analysis, Univ. La Laguna, Tenerife, Canary Islands, Spain
Email: pglez@ull.es

R. Orive
Affiliation: Department Mathematical Analysis, Univ. La Laguna, Tenerife, Canary Islands, Spain
Email: rorive@ull.es

DOI: http://dx.doi.org/10.1090/S0025-5718-99-01107-2
PII: S 0025-5718(99)01107-2
Received by editor(s): March 3, 1998
Received by editor(s) in revised form: May 19, 1998
Published electronically: February 24, 1999
Additional Notes: The work of the first author is partially supported by the Fund for Scientific Research (FWO), project “Orthogonal systems and their applications”, grant #G.0278.97, and the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister’s Office for Science, Technology and Culture. The scientific responsibility rests with the author.
The work of the other three authors was partially supported by the scientific research project PB96-1029 of the Spanish D.G.I.C.Y.T
Dedicated: Dedicated to Professor Nácere Hayek Calil on the occasion of his 75th birthday
Article copyright: © Copyright 2000 American Mathematical Society