Error bounds of Gaussian quadrature formulae for one class of Bernstein-Szegő weights

Author:
Miodrag M. Spalević

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

Published electronically:
December 14, 2012

MathSciNet review:
3008848

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The kernels in the remainder terms of the Gaussian quadrature formulae for analytic functions inside elliptical contours with foci at and a sum of semi-axes , when the weight function is of Bernstein-Szegő type

**1.**W. Gautschi and S. E. Notaris,*Gauss-Kronrod quadrature formulae for weight function of Bernstein-Szegő type*, J. Comput. Appl. Math.**25**(1989), 199-224; erratum in J. Comput. Appl. Math.**27**(1989), 429. MR**988057 (90d:65045)****2.**W. Gautschi and R. S. Varga,*Error bounds for Gaussian quadrature of analytic functions*, SIAM J. Numer. Anal.**20**(1983), 1170-1186. MR**723834 (85j:65010)****3.**W. Gautschi, E. Tychopoulos and R. S. Varga,*A note on the contour integral representation of the remainder term for a Gauss-Chebyshev quadrature rule*, SIAM J. Numer. Anal.**27**(1990), 219-224. MR**1034931 (91d:65044)****4.**D. B. Hunter,*Some error expansions for Gaussian quadrature*, BIT**35**(1995), 64-82. MR**1429008 (97i:65040)****5.**S. E. Notaris,*The error norm of Gaussian quadrature formulae for weight functions of Bernstein-Szegő type*, Numer. Math.**57**(1990), 271-283. MR**1057125 (92a:65080)****6.**F. Peherstorfer,*On the remainder of Gaussian quadrature formulas for Bernstein-Szegő weight functions*, Math. Comp.**60**(1993), 317-325. MR**1153169 (93d:65030)****7.**R. Scherer, T. Schira,*Estimating quadrature errors for analytic functions using kernel representations and biorthogonal systems*, Numer. Math.**84**(2000), 497-518. MR**1738478 (2001d:65034)****8.**T. Schira,*The remainder term for analytic functions of symmetric Gaussian quadratures*, Math. Comp.**66**(1997), 297-310. MR**1372009 (97c:65050)****9.**M. M. Spalević and M. S. Pranić,*Error bounds of certain Gaussian quadrature formulae*,

J. Comput. Appl. Math.**234**(2010), 1049-1057. MR**2609559 (2011b:41047)****10.**M. M. Spalević, M. S. Pranić and A. V. Pejčev,*Maximum of the modulus of kernels of Gaussian quadrature formulae for one class of Bernstein-Szegő weight functions*, Appl. Math. Comput.**218**(2012), 5746-5756.**11.**F. Stenger,*Bounds on the error of Gauss-type quadratures*, Numer. Math.**8**(1966), 150-160. MR**0196936 (33:5120)****12.**B. von Sydow,*Error estimates for Gaussian quadrature formulae*, Numer. Math.**29**(1977), 59-64. MR**0471270 (57:11007)**

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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

Email:
mspalevic@mas.bg.ac.rs

DOI:
https://doi.org/10.1090/S0025-5718-2012-02667-6

Keywords:
Maximum,
modulus,
kernel,
Gaussian quadrature formula,
Bernstein-Szegő weight function

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)

Article copyright:
© Copyright 2012
American Mathematical Society