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An Error Expansion for some Gauss–Turán Quadratures and L1-Estimates of the Remainder Term

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Abstract

Our aim in this paper is to obtain error expansions in the Gauss–Turán quadrature formula ∫ 1−1 f(t)w(t) dt=∑ nν=1 ∑ 2si=0 Ai,νf(i)(τν)+Rn,s(f), in the case when f is an analytic function in some region of the complex plane containing the interval [−1,1] in its interior. Using a representation of the remainder term Rn,s(f) in the form of contour integral over confocal ellipses, we obtain Rn,1(f) for the four Chebyshev weights and Rn,2(f) for the Chebyshev weight of the first kind. Also, we get a few new L1-estimates of the remainder term, which are stronger than the previous ones. Some numerical results, illustrations and comparisons are also given.

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Authors and Affiliations

  1. Department Mathematics, University of Niš, Faculty of Electronic Engineering, P.O. Box 73, 18000, Niš, Serbia

    G. V. Milovanović

  2. Department of Mathematics and Informatics, University of Kragujevac, Faculty of Science, P.O. Box 60, 34000, Kragujevac, Serbia

    M. M. Spalević

Authors
  1. G. V. Milovanović
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  2. M. M. Spalević
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Corresponding author

Correspondence to G. V. Milovanović.

Additional information

AMS subject classification (2000)

41A55, 65D30, 65D32.

Received January 2004. Accepted October 2004. Communicated by Lothar Reichel.

M. M. Spalević: This work was supported in part by the Serbian Ministry of Science and Environmental Protection (Project: Applied Orthogonal Systems, Constructive Approximation and Numerical Methods, grant number 2002).

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Milovanović, G.V., Spalević, M.M. An Error Expansion for some Gauss–Turán Quadratures and L1-Estimates of the Remainder Term. Bit Numer Math 45, 117–136 (2005). https://doi.org/10.1007/s10543-005-2643-y

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