Summary
We consider Gaussian quadrature formulaeQ n , n∈ℕ, approximating the integral\(I(f): = \int\limits_{ - 1}^1 {w(x)f(x)dx} \), wherew is a weight function. In certain spaces of analytic functions the error functionalR n :=I−Q n is continuous. Previously one of the authors deduced estimates for ‖R n ‖ for symmetric Gaussian quadrature formulae. In this paper we extend these results to nonsymmetric Gaussian formulae using a recent result of Gautschi concerning the sign ofR n (q K ),q K (x):=x K, for a wide class of weight functions including the Jacobi weights.
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Akrivis, G.: Fehlerabschätzungen bei der numerischen Integration in einer und mehreren Dimensionen. Dissertation, Universität München (1983)
Akrivis, G.: Fehlerabschätzungen für Gauß-Quadraturformeln. Numer. Math.44, 261–278 (1984)
Braß, H.: Quadraturverfahren. Göttingen, Zürich: Vandenhoeck und Ruprecht 1977
Chawla, M.M.: Hilbert space for estimating errors of quadratures for analytic functions. BIT10, 145–155 (1970)
Davis, P.J., Rabinowitz, P.: Methods of numerical integration. New York, San Francisco, London Academic Press 1975
Gautschi, W.: On Padé approximants associated with Hamburger series. Calcolo20, 111–127 (1983)
Gautschi, W., Varga, R.S.: Error bounds for Gaussian quadrature of analytic functions. SIAM J. Numer. Anal.20, 1170–1186 (1983)
Hämmerlin, G.: Fehlerabschätzungen bei numerischer Integration nach Gauß In: Methoden und Verfahren der mathematischen Physik, Bd. 6, (Hrsg. B. Brosowski, E. Martensen), S. 153–163, Mannheim, Wien, Zürich: Bibliographisches Institut 1972
Rivlin, T.J.: Polynominals of best uniform approximation to certain rational functions. Numer. Math.4, 345–349 (1962)
Szegö, G.: Orthogonal polynomials. New York: Amer. Math. Soc. 1939
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Akrivis, G., Burgstaller, A. Fehlerabschätzungen für nichtsymmetrische Gauß-Quadraturformeln. Numer. Math. 47, 535–543 (1985). https://doi.org/10.1007/BF01389455
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DOI: https://doi.org/10.1007/BF01389455