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A Quadrature Formula of Infinite Order

  • R. Gervais
  • Q. I. Rahman
  • G. Schmeisser
Chapter
Part of the ISNM 57: International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique book series (ISNM, volume 57)

Abstract

Given a quadrature formula
$$\int\limits_{ - 1}^1 {f(x)dx = \sum\limits_{\nu = 1}^n {{A_\nu }f({x_\nu }) + R\left[ f \right]} } $$
(1)
there exists a largest non-negative integer k such that R[p] vanishes for every polynomial p of degree less than k. This number k is usually called the order and k-1 the degree of precision of the formula (1). It is well-known that k ≤ 2n where equality is attained for the Gauss-Legendre formula only.

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References

  1. 1.
    Beurling, A. and Malliavin, P. On Fourier transforms of measures with compact support. Acta Math. 107 (1962), 291–309.CrossRefGoogle Scholar
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    Boas, R.P. Entire functions. Academic Press, New York 1954.Google Scholar
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    Mc Namee, J., Stenger, F. and Whitney, E.L. Whittaker’s cardinal function in retrospect. Math. Comp. 25 (1971), 141–154.Google Scholar
  4. 4.
    Timan, A.F. Theory of approximation of a real variable. Pergamon Press, Oxford-London-New York-Paris 1963.Google Scholar

Copyright information

© Springer Basel AG 1982

Authors and Affiliations

  • R. Gervais
  • Q. I. Rahman
  • G. Schmeisser

There are no affiliations available

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