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

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- by A. Bultheel, C. Díaz-Mendoza, P. González-Vera and R. Orive;
- Math. Comp.
**69**(2000), 721-747 - DOI: https://doi.org/10.1090/S0025-5718-99-01107-2
- Published electronically: February 24, 1999
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## 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.## References

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## Bibliographic 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
- 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 - © Copyright 2000 American Mathematical Society
- Journal: Math. Comp.
**69**(2000), 721-747 - MSC (1991): Primary 65D30; Secondary 41A21
- DOI: https://doi.org/10.1090/S0025-5718-99-01107-2
- MathSciNet review: 1651743

Dedicated: Dedicated to Professor Nácere Hayek Calil on the occasion of his 75th birthday