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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.

References [Enhancements On Off] (What's this?)

  • 1. M. Abramowitz and A. Stegun, Handbook of mathematical functions, Dover Publications, New York, 1966. MR 34:8606
  • 2. A. Bultheel, C. Díaz-Mendoza, P. González-Vera, and R. Orive, Quadrature on the half line and two-point Padé approximants to Stieltjes functions. Part II: Convergence, J. Comput. Appl. Math. 77 (1997), 53-76. MR 98a:41023
  • 3. -, Quadrature on the half line and two-point Padé approximants to Stieltjes functions. Part III: The unbounded case, J. Comput. Appl. Math. 87 (1997), 95-117. MR 98j:41033
  • 4. A. Bultheel, P. González-Vera, and R. Orive, Quadrature on the half line and two-point Padé approximants to Stieltjes functions. Part I: Algebraic aspects, J. Comput. Appl. Math. 65 (1995), 57-72. MR 96m:41018
  • 5. T. Chihara, An introduction to orthogonal polynomials, Gordon and Breach Science Publishers, New York, 1978. MR 58:1979
  • 6. L. Cochran and S.C. Cooper, Orthogonal Laurent polynomials on the real line, Continued fractions and orthogonal functions (New York) (S.C. Cooper and W.J. Thron, eds.), Marcel Dekker, 1994, pp. 47-100. MR 95b:42024
  • 7. P.J. Davis and P. Rabinowitz, Methods of numerical integration, 2nd ed., Academic Press, 1984. MR 86d:65004
  • 8. A. Erdelyi (ed.), Higher transcendental functions, vol. 3, Mc Graw-Hill, New York, 1955. MR 16:586c
  • 9. W. Gautschi, A survey of Gauss-Christoffel quadrature formulae, E.B. Christoffel. The influence of his work on mathematical and physical sciences (Basel) (P.L. Butzer and F. Feher, eds.), Birkhäuser Verlag, 1981, pp. 72-147. MR 83g:41031
  • 10. C. González-Concepción, P. González-Vera, and L. Casasús, On the convergence of certain quadrature formulas defined on unbounded intervals, Orthogonal Polynomials and their Applications (J. Vinuesa, ed.), Lecture Notes in Pure and Appl. Math., vol. 117, Marcel Dekker Inc., 1989, pp. 147-151. MR 91f:41037
  • 11. W.B. Jones, O. Njåstad, and W.J. Thron, Two-point Padé expansions for a family of analytic functions, J. Comput. Appl. Math. 9 (1983), 105-124. MR 84j:30057
  • 12. W.B. Jones and W.J. Thron, Orthogonal Laurent polynomials and Gaussian quadrature, Quantum mechanics in mathematics, chemistry and physics (New York) (K. Gustafson and W.P. Reinhardt, eds.), Plenum, 1984, pp. 449-455.
  • 13. W.B. Jones, W.J. Thron, and H. Waadeland, A strong Stieltjes moment problem, Trans. Amer. Math. Soc. 206 (1980), 503-528. MR 81j:30055
  • 14. V.J. Krylov, Approximate calculation of integrals, MacMillan, 1962. MR 26:2008
  • 15. G. López-Lagomasino and A. Martínez-Finkelshtein, Rate of convergence of two-point Padé approximants and logarithmic asymptotics of Laurent-type orthogonal polynomials, Constr. Approx. 11 (1995), 255-286. MR 96i:41016
  • 16. G.L. Lopes [López-Lagomasino], On the convergence of Padé approximants for Stieltjes type functions, Math. USSR-Sb. 39 (1981), 281-288. MR 81m:30034
  • 17. -, On the asymptotics of the ratio of orthogonal polynomials and convergence of multipoint Padé approximants, Math. USSR-Sb. 56 (1985), 207-219. MR 87e:30050
  • 18. -, Convergence of Padé approximants of Stieltjes type meromorphic functions and comparative asymptotice for orthogonal polynomials, Math. USSR-Sb. 64 (1989), 207-227. MR 90g:30003
  • 19. A. Sri Ranga, Another quadrature rule of highest algebraic degree, Numer. Math. 28 (1994), 283-294. MR 95c:65047
  • 20. A. Sri Ranga and J.H. McCabe, On the extensions of some classical distributions, Proc. Edinburgh Math. Soc. 34 (1991), 12-29. MR 92b:30003
  • 21. T.J. Stieltjes, Quelques recherches sur la théorie des quadratures dites mécaniques, Ann. Acad. École Norm. Paris, Sér. 3 1 (1884), 409-426, Oeuvres vol. 1, pp. 377-396. MR 95g:01033
  • 22. -, Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse 8 (1894), J.1-122, 9:1-47, Enlish transl.: Oeuvres Complèts, Collected Papers, Springer Verlag, 1993, Vol. 2, 609-745. MR 95g:01033
  • 23. J.V. Uspensky, On the convergence of quadrature formulas between infinite limits, Bulletin of the Russian Academy of Sciences (1916).
  • 24. -, On the convergence of quadrature formulas related to an infinite interval, Trans. Amer. Math. Soc. 30 (1928), 542-554.

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Additional Information

A. Bultheel
Affiliation: Department of Computer Science, K.U. Leuven, Celestijnenlaan 200A, B-3001 Heverlee, Belgium

C. Díaz-Mendoza
Affiliation: Department Mathematical Analysis, Univ. La Laguna, Tenerife, Canary Islands, Spain

P. González-Vera
Affiliation: Department Mathematical Analysis, Univ. La Laguna, Tenerife, Canary Islands, Spain

R. Orive
Affiliation: Department Mathematical Analysis, Univ. La Laguna, Tenerife, Canary Islands, Spain

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

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