Summary
In certain spaces of analytic functions the error term of the Gauss-Lobatto quadrature formula relative to a (nonnegative) weight function is a continuous linear functional. Here we compute the norm of the error functional for the Bernstein-Szegö weight functions consisting of any of the four Chebyshev weights divided by an arbitrary quadratic polynomial that remains positive on [−1, 1]. The norm can subsequently be used to derive bounds for the error functional. The efficiency of these bounds is illustrated with some numerical examples.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Akrivis, G., Burgstaller, A. (1985): Fehlerabschätzungen für nichtsymmetrische Gauss-Quadraturformeln. Numer. Math.47, 535–543
Gautschi, W. (1981): A survey of Gauss-Christoffel quadrature formulae. In: P.L. Butzer, F. Fehér, eds., E.B. Christoffel, pp. 72–147. Birkhäuser, Basel
Gautschi, W. (1991): On the remainder term for analytic functions of Gauss-Lobatto and Gauss-Radau quadratures. Rocky Mountain J. Math.21, 209–226
Gautschi, W., Notaris, S.E. (1989): Gauss-Kronrod quadrature formulae for weight functions of Bernstein-Szegö type. J. Comput. Appl. Math.25, 199–224; erratum in: J. Comput. Appl. Math.27, 429
Gautschi, W., Varga, R.S. (1983): Error bounds for Gaussian quadrature of analytic functions. SIAM J. Numer. Anal.20, 1170–1186
Notaris, S.E. (1990): The error norm of Gaussian quadrature formulae for weight functions of Bernstein-Szegö type. Numer. Math.57, 271–283
Hämmerlin, G. (1972): Fehlerabschätzung bei numerischer Integration nach Gauss. In: B. Brosowski, E. Martensen, eds., Methoden und Verfahren der mathematischen Physik, vol. 6, pp. 153–163. Bibliographisches Institut, Mannheim Wien Zürich
Rivlin, T.J. (1990): Chebyshev Polynomials, 2nd ed. Wiley, New York
Szegö, G. (1975): Orthogonal Polynomials. Colloquium Publications, vol. 23, 4th ed. American Mathematical Society, Providence, R.I.
Author information
Authors and Affiliations
Additional information
Work supported in part by a grant from the Research Council of the Graduate School, University of Missouri-Columbia.
Rights and permissions
About this article
Cite this article
Notaris, S.E. The error norm of Gauss-Lobatto quadrature formulae for weight functions of Bernstein-Szegö type. Numer. Math. 64, 381–393 (1993). https://doi.org/10.1007/BF01388695
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01388695