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Numerical Algorithms

, Volume 77, Issue 4, pp 1003–1028 | Cite as

The error bounds of Gauss-Kronrod quadrature formulae for weight functions of Bernstein-Szegő type

  • Dušan Lj. Djukić
  • Aleksandar V. Pejčev
  • Miodrag M. Spalević
Original Paper
  • 63 Downloads

Abstract

We consider the Gauss-Kronrod quadrature formulae for the Bernstein-Szegő weight functions consisting of any one of the four Chebyshev weights divided by the polynomial \(\rho (t)=1-\frac {4\gamma }{(1+\gamma )^{2}}\,t^{2},\quad t\in (-1,1),\ -1<\gamma \le 0\). For analytic functions, the remainder term of this quadrature formula can be represented as a contour integral with a complex kernel. We study the kernel, on elliptic contours with foci at the points ∓ 1 and sum of semi-axes ρ > 1, for the given quadrature formula. Starting from the explicit expression of the kernel, we determine the locations on the ellipses where maximum modulus of the kernel is attained. So we derive effective error bounds for this quadrature formula. An alternative approach, which has initiated this research, has been proposed by S. Notaris (Numer. Math. 103, 99–127, 2006).

Keywords

Gauss-Kronrod quadrature formulae Bernstein-Szegő weight functions Contour integral representation Remainder term for analytic functions Error bound 

Mathematics Subject Classification (2010)

65D32 

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Notes

Acknowledgements

We are indebted to the referees for making suggestions that have improved the paper.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Dušan Lj. Djukić
    • 1
  • Aleksandar V. Pejčev
    • 1
  • Miodrag M. Spalević
    • 1
  1. 1.Department of Mathematics, Faculty of Mechanical EngineeringUniversity of BeogradBelgradeSerbia

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