The error norm of Gauss-Radau quadrature formulae for Bernstein-Szegö weight functions
HTML articles powered by AMS MathViewer
- by Sotirios E. Notaris PDF
- Math. Comp. 84 (2015), 2843-2865 Request permission
Abstract:
We consider the Gauss-Radau quadrature formulae for the Bernstein-Szegö weight functions consisting of any one of the four Chebyshev weights divided by the polynomial $\rho (t)=1-\frac {4\gamma }{(1+\gamma )^{2}}t^{2},\ -1<t<1,\ -1<\gamma \leq 0$. On certain spaces of analytic functions the error term of these formulae is a continuous linear functional. We compute or estimate the norm of the error functional.References
- H. Engels, Numerical quadrature and cubature, Computational Mathematics and Applications, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1980. MR 587486
- Walter Gautschi, A survey of Gauss-Christoffel quadrature formulae, E. B. Christoffel (Aachen/Monschau, 1979) Birkhäuser, Basel-Boston, Mass., 1981, pp. 72–147. MR 661060
- Walter Gautschi, Remainder estimates for analytic functions, Numerical integration (Bergen, 1991) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 357, Kluwer Acad. Publ., Dordrecht, 1992, pp. 133–145. MR 1198903
- W. Gautschi, “Algorithm 726: ORTHPOL - a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules”, ACM Trans. Math. Software, 20 (1994), no. 1, 21-62; “Remark on algorithm 726: ORTHPOL - a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules”, ACM Trans. Math. Software, 24 (1998), no. 3, 355-356.
- Walter Gautschi, Orthogonal polynomials: computation and approximation, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2004. Oxford Science Publications. MR 2061539
- Walter Gautschi and Sotorios E. Notaris, Gauss-Kronrod quadrature formulae for weight functions of Bernstein-Szegő type, J. Comput. Appl. Math. 25 (1989), no. 2, 199–224. MR 988057, DOI 10.1016/0377-0427(89)90047-2
- I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London-Toronto, Ont., 1980. Corrected and enlarged edition edited by Alan Jeffrey; Incorporating the fourth edition edited by Yu. V. Geronimus [Yu. V. Geronimus] and M. Yu. Tseytlin [M. Yu. Tseĭtlin]; Translated from the Russian. MR 582453
- Sotirios E. Notaris, The error norm of Gaussian quadrature formulae for weight functions of Bernstein-Szegő type, Numer. Math. 57 (1990), no. 3, 271–283. MR 1057125, DOI 10.1007/BF01386411
- Sotirios E. Notaris, The error norm of Gauss-Kronrod quadrature formulae for weight functions of Bernstein-Szegő type, Numer. Math. 103 (2006), no. 1, 99–127. MR 2207616, DOI 10.1007/s00211-005-0670-x
- Sotirios E. Notaris, The error norm of Gauss-Radau quadrature formulae for Chebyshev weight functions, BIT 50 (2010), no. 1, 123–147. MR 2595479, DOI 10.1007/s10543-010-0252-x
- Sotirios E. Notaris, The error norm of quadrature formulae, Numer. Algorithms 60 (2012), no. 4, 555–578. MR 2946482, DOI 10.1007/s11075-012-9582-x
- Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. MR 0372517
Additional Information
- Sotirios E. Notaris
- Affiliation: Department of Mathematics, University of Athens, Panepistemiopolis, 15784 Athens, Greece
- Email: notaris@math.uoa.gr
- Received by editor(s): August 6, 2013
- Received by editor(s) in revised form: February 1, 2014
- Published electronically: April 15, 2015
- © Copyright 2015 American Mathematical Society
- Journal: Math. Comp. 84 (2015), 2843-2865
- MSC (2010): Primary 65D32
- DOI: https://doi.org/10.1090/mcom/2944
- MathSciNet review: 3378850
Dedicated: Dedicated to Walter Gautschi on his 85th birthday with respect and admiration