Error bounds of Gaussian quadrature formulae for one class of Bernstein-Szegő weights
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- by Miodrag M. Spalević PDF
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Abstract:
The kernels $K_n(z)$ in the remainder terms $R_n(f)$ of the Gaussian quadrature formulae for analytic functions $f$ inside elliptical contours with foci at $\mp 1$ and a sum of semi-axes $\rho >1$, when the weight function $w$ is of Bernstein-Szegő type \[ w(t)\equiv w_\gamma ^{(-1/2,1/2)}(t)=\displaystyle \sqrt {\frac {1+t}{1-t}}\cdot \frac {1}{1-\displaystyle \frac {4\gamma }{(1+\gamma )^2} t^2}, \quad t\in (-1,1),\quad \gamma \in (-1,0), \] are studied. Sufficient conditions are found ensuring that the kernel attains its maximal absolute value at the intersection point of the contour with the positive real semi-axis. This leads to effective error bounds of the corresponding Gauss quadratures. The quality of the derived bounds is analyzed by a comparison with other error bounds intended for the same class of integrands. In part our analysis is based on the well-known Cardano formulas, which are not very popular among mathematicians.References
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Additional Information
- Miodrag M. Spalević
- Affiliation: Department of Mathematics, University of Beograd, Faculty of Mechanical Engineering, Kraljice Marije 16, 11120 Belgrade 35, Serbia
- MR Author ID: 600543
- Email: mspalevic@mas.bg.ac.rs
- Received by editor(s): September 22, 2010
- Received by editor(s) in revised form: March 6, 2011
- Published electronically: December 14, 2012
- Additional Notes: The author was supported in part by the Serbian Ministry of Education and Science (Research Project: “Methods of numerical and nonlinear analysis with applications”, No. #174002)
- © Copyright 2012 American Mathematical Society
- Journal: Math. Comp. 82 (2013), 1037-1056
- MSC (2010): Primary 41A55; Secondary 65D30, 65D32
- DOI: https://doi.org/10.1090/S0025-5718-2012-02667-6
- MathSciNet review: 3008848