Maximum of the modulus of kernels of Gaussian quadrature formulae for one class of Bernstein-Szego weight functions
Abstract
We continue with the study of 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 -/+ 1 and a sum of semi-axes rho > 1. The weight function w of Bernstein-Szego type here is w(t) equivalent to w(gamma)((-1/2))(t) = 1/root 1 - t(2) . (1 - 4 gamma/(1 + gamma)(2)t(2))(-1), t is an element of (-1, 1), gamma is an element of (-1, 0). Sufficient conditions are found ensuring that the kernel attains its maximum absolute value at the intersection point of the contour with either the real or the imaginary axis. This leads to effective error bounds of the corresponding Gauss quadratures. The quality of the derived bounds is demonstrated by a comparison with other error bounds intended for the same class of integrands.
Keywords:
Remainder term / Kernel / Gauss quadrature / Error bound / Elliptic contour / Analytic functionSource:
Applied Mathematics and Computation, 2012, 218, 9, 5746-5756Publisher:
- Elsevier Science Inc, New York
Funding / projects:
- Methods of Numerical and Nonlinear Analysis with Applications (RS-MESTD-Basic Research (BR or ON)-174002)
DOI: 10.1016/j.amc.2011.11.072
ISSN: 0096-3003
WoS: 000298293200096
Scopus: 2-s2.0-83555166266
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Mašinski fakultetTY - JOUR AU - Spalević, Miodrag AU - Pranić, Miroslav S. AU - Pejčev, Aleksandar PY - 2012 UR - https://machinery.mas.bg.ac.rs/handle/123456789/1388 AB - We continue with the study of 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 -/+ 1 and a sum of semi-axes rho > 1. The weight function w of Bernstein-Szego type here is w(t) equivalent to w(gamma)((-1/2))(t) = 1/root 1 - t(2) . (1 - 4 gamma/(1 + gamma)(2)t(2))(-1), t is an element of (-1, 1), gamma is an element of (-1, 0). Sufficient conditions are found ensuring that the kernel attains its maximum absolute value at the intersection point of the contour with either the real or the imaginary axis. This leads to effective error bounds of the corresponding Gauss quadratures. The quality of the derived bounds is demonstrated by a comparison with other error bounds intended for the same class of integrands. PB - Elsevier Science Inc, New York T2 - Applied Mathematics and Computation T1 - Maximum of the modulus of kernels of Gaussian quadrature formulae for one class of Bernstein-Szego weight functions EP - 5756 IS - 9 SP - 5746 VL - 218 DO - 10.1016/j.amc.2011.11.072 ER -
@article{ author = "Spalević, Miodrag and Pranić, Miroslav S. and Pejčev, Aleksandar", year = "2012", abstract = "We continue with the study of 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 -/+ 1 and a sum of semi-axes rho > 1. The weight function w of Bernstein-Szego type here is w(t) equivalent to w(gamma)((-1/2))(t) = 1/root 1 - t(2) . (1 - 4 gamma/(1 + gamma)(2)t(2))(-1), t is an element of (-1, 1), gamma is an element of (-1, 0). Sufficient conditions are found ensuring that the kernel attains its maximum absolute value at the intersection point of the contour with either the real or the imaginary axis. This leads to effective error bounds of the corresponding Gauss quadratures. The quality of the derived bounds is demonstrated by a comparison with other error bounds intended for the same class of integrands.", publisher = "Elsevier Science Inc, New York", journal = "Applied Mathematics and Computation", title = "Maximum of the modulus of kernels of Gaussian quadrature formulae for one class of Bernstein-Szego weight functions", pages = "5756-5746", number = "9", volume = "218", doi = "10.1016/j.amc.2011.11.072" }
Spalević, M., Pranić, M. S.,& Pejčev, A.. (2012). Maximum of the modulus of kernels of Gaussian quadrature formulae for one class of Bernstein-Szego weight functions. in Applied Mathematics and Computation Elsevier Science Inc, New York., 218(9), 5746-5756. https://doi.org/10.1016/j.amc.2011.11.072
Spalević M, Pranić MS, Pejčev A. Maximum of the modulus of kernels of Gaussian quadrature formulae for one class of Bernstein-Szego weight functions. in Applied Mathematics and Computation. 2012;218(9):5746-5756. doi:10.1016/j.amc.2011.11.072 .
Spalević, Miodrag, Pranić, Miroslav S., Pejčev, Aleksandar, "Maximum of the modulus of kernels of Gaussian quadrature formulae for one class of Bernstein-Szego weight functions" in Applied Mathematics and Computation, 218, no. 9 (2012):5746-5756, https://doi.org/10.1016/j.amc.2011.11.072 . .