Summary
Due to a theorem of Peano, the remainder term of Gregory's quadrature formula may be expressed by means of a polynomial spline-function. An explicite representation of this spline-function is given by a direct method, which does not make use of Peano's theorem. As an application the remainder term obtained is developed in terms of boundary derivations of the integrand, which leads to a general quadrature formula containing the Gregory formula and the Euler-MacLaurin summation formula as special cases.
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Literatur
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Herrn Johannes Weissinger zum 60. Geburtstag am 12. 5. 1973 gewidmet
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Martensen, E. Darstellung und Entwicklung des Restgliedes der Gregoryschen Quadraturformel mit Hilfe von Spline-Funktionen. Numer. Math. 21, 70–80 (1973). https://doi.org/10.1007/BF01436188
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DOI: https://doi.org/10.1007/BF01436188