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Numerische Mathematik

, Volume 132, Issue 1, pp 201–216 | Cite as

Euler–Maclaurin and Gregory interpolants

  • Mohsin Javed
  • Lloyd N. TrefethenEmail author
Article

Abstract

Let a sufficiently smooth function \(f\) on \([-1,1]\) be sampled at \(n+1\) equispaced points, and let \(k\ge 0\) be given. An Euler–Maclaurin interpolant to the data is defined, consisting of a sum of a degree \(k\) algebraic polynomial and a degree \(n\) trigonometric polynomial, which deviates from \(f\) by \(O(n^{-k})\) and whose integral is equal to the order \(k\) Euler–Maclaurin approximation of the integral of \(f\). This interpolant makes use of the same derivatives \(f^{( j)}(\pm 1)\) as the Euler–Maclaurin formula. A variant Gregory interpolant is also defined, based on finite difference approximations to the derivatives, whose integral (for \(k\) odd) is equal to the order \(k\) Gregory approximation to the integral.

Mathematics Subject Classification

41A05 42A15 65D32 65D05 

Notes

Acknowledgments

We have benefited from helpful remarks from Jean-Paul Berrut, Walter Gautschi, Kai Hormann, Rodrigo Platte, Jared Tanner, and Grady Wright. We are also grateful to Bengt Fornberg for drawing our attention to the power and simplicity of Gregory quadrature. The second author thanks Martin Gander of the University of Geneva for hosting a sabbatical visit during which this article was written, just down the street from Darbes’ memorable oil painting of Euler in the Musée d’art et d’histoire.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of OxfordOxfordUK

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