Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Maximum of the modulus of kernels in Gauss-Turán quadratures

Authors: Gradimir V. Milovanovic, Miodrag M. Spalevic and Miroslav S. Pranic
Journal: Math. Comp. 77 (2008), 985-994
MSC (1991): Primary 41A55; Secondary 65D30, 65D32
Published electronically: November 14, 2007
MathSciNet review: 2373188
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the kernels $ K_{n,s}(z)$ in the remainder terms $ R_{n,s}(f)$ of the Gauss-Turán quadrature formulae for analytic functions on elliptical contours with foci at $ \pm 1$, when the weight $ \omega$ is a generalized Chebyshev weight function. For the generalized Chebyshev weight of the first (third) kind, it is shown that the modulus of the kernel $ \vert K_{n,s}(z)\vert$ attains its maximum on the real axis (positive real semi-axis) for each $ n\geq n_0, n_0=n_0(\rho,s)$. It was stated as a conjecture in [Math. Comp. 72 (2003), 1855-1872]. For the generalized Chebyshev weight of the second kind, in the case when the number of the nodes $ n$ in the corresponding Gauss-Turán quadrature formula is even, it is shown that the modulus of the kernel attains its maximum on the imaginary axis for each $ n\geq n_0, n_0=n_0(\rho,s)$. Numerical examples are included.

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

  • 1. S. Bernstein, Sur les polynomes orthogonaux relatifs à un segment fini, J. Math. Pures Appl. 9 (1930), 127-177.
  • 2. J. D. Donaldson and D. Elliott, A unified approach to quadrature rules with asymptotic estimates of their remainders, SIAM J. Numer. Anal. 9 (1972), 573-602. MR 0317522 (47:6069)
  • 3. W. Gautschi and G. V. Milovanovic, $ S$-orthogonality and construction of Gauss-Turán type quadrature formulae, J. Comput. Appl. Math. 86 (1997), 205-218. MR 1491435 (99a:65030)
  • 4. W. Gautschi and S. E. Notaris, Gauss-Kronrod quadrature formulae for weight function of Bernstein-Szegö type, J. Comput. Appl. Math. 25 (1989), 199-224. MR 988057 (90d:65045)
  • 5. W. Gautschi and R. S. Varga, Error bounds for Gaussian quadrature of analytic functions, SIAM J. Numer. Anal. 20 (1983), 1170-1186. MR 723834 (85j:65010)
  • 6. W. Gautschi, E. Tychopoulos and R. S. Varga, A note on the contour integral representation of the remainder term for a Gauss-Chebyshev quadrature rule, SIAM J. Numer. Anal. 27 (1990), 219-224. MR 1034931 (91d:65044)
  • 7. A. Ghizzetti and A. Ossicini, Quadrature formulae, Akademie - Verlag, Berlin, 1970. MR 0269116 (42:4012)
  • 8. D. B. Hunter, G. Nikolov, On the error term of symmetric Gauss-Lobatto quadrature formulae for analytic functions, Math. Comp. 69 (2000), 269-282. MR 1642754 (2001a:65030)
  • 9. G. V. Milovanovic, Quadratures with multiple nodes, power orthogonality, and moment-preserving spline approximation, in Numerical Analysis in 20th Century, Vol. 5 (W. Gautschi, F. Marcellán, and L. Reichel, eds.), J. Comput. Appl. Math. 127 (2001), 267-286. MR 1808578 (2002e:65039)
  • 10. G. V. Milovanovic and M. M. Spalevic, Quadrature formulae connected to $ \sigma$-orthogonal polynomials, J. Comput. Appl. Math. 140 (2002), 619-637. MR 1934463 (2003h:65025)
  • 11. G. V. Milovanovic and M. M. Spalevic, Error bounds for Gauss-Turán quadrature formulae of analytic functions, Math. Comp. 72 (2003), 1855-1872. MR 1986808 (2004c:41068)
  • 12. G. V. Milovanovic, M. M. Spalevic, An error expansion for Gauss-Turán quadratures and $ L^1$-estimates of the remainder term, BIT 45 (2005), 117-136. MR 2164228 (2006e:41058)
  • 13. G. V. Milovanovic, M. M. Spalevic and A. S. Cvetkovic, Calculation of Gaussian quadratures with multiple nodes, Math. Comput. Modelling 39 (2004), 325-347. MR 2037399 (2005g:65041)
  • 14. A. Ossicini and F. Rosati, Funzioni caratteristiche nelle formule di quadratura gaussiane con nodi multipli, Boll. Un. Mat. Ital. (4) 11 (1975), 224-237. MR 0408212 (53:11977)
  • 15. T. Schira, The remainder term for analytic functions of Gauss-Lobatto quadratures, J. Comput. Appl. Math. 76 (1996), 171-193. MR 1423516 (97m:41033)
  • 16. T. Schira, The remainder term for analytic functions of symmetric Gaussian quadratures, Math. Comp. 66 (1997), 297-310. MR 1372009 (97c:65050)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (1991): 41A55, 65D30, 65D32

Retrieve articles in all journals with MSC (1991): 41A55, 65D30, 65D32

Additional Information

Gradimir V. Milovanovic
Affiliation: Department of Mathematics, University of Niš, Faculty of Electronic Engineering, P.O. Box 73, 18000 Niš, Serbia

Miodrag M. Spalevic
Affiliation: Department of Mathematics and Informatics, University of Kragujevac, Faculty of Science, P.O. Box 60, 34000 Kragujevac, Serbia

Miroslav S. Pranic
Affiliation: Department of Mathematics and Informatics, University of Banja Luka, Faculty of Science, M. Stojanovića 2, 51000 Banja Luka, Bosnia and Herzegovina

Keywords: Gauss-Tur\'an quadrature, Chebyshev weight functions, remainder term for analytic functions, error estimate, contour integral representation, confocal ellipses, kernel.
Received by editor(s): August 15, 2006
Received by editor(s) in revised form: December 4, 2006
Published electronically: November 14, 2007
Additional Notes: The authors were supported in part by the Swiss National Science Foundation (SCOPES Joint Research Project No. IB7320-111079 “New Methods for Quadrature”) and the Serbian Ministry of Science (Research Projects: “Approximation of linear operators” (No. #144005) & “Orthogonal systems and applications” (No. #144004C))
Article copyright: © Copyright 2007 American Mathematical Society

American Mathematical Society