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

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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).

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Acknowledgements

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

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Mechanical Engineering, University of Beograd, Kraljice Marije 16, 11120 Belgrade 35, Belgrade, Serbia

    Dušan Lj. Djukić, Aleksandar V. Pejčev & Miodrag M. Spalević

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  1. Dušan Lj. Djukić
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  2. Aleksandar V. Pejčev
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  3. Miodrag M. Spalević
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Correspondence to Dušan Lj. Djukić.

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The authors were supported in part by the Serbian Ministry of Education, Science and Technological Development (Research Project: “Methods of numerical and nonlinear analysis with applications” (No. #174002)).

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Djukić, D.L., Pejčev, A.V. & Spalević, M.M. The error bounds of Gauss-Kronrod quadrature formulae for weight functions of Bernstein-Szegő type. Numer Algor 77, 1003–1028 (2018). https://doi.org/10.1007/s11075-017-0351-8

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