Riemann sums over polytopes
Annales de l'Institut Fourier, Volume 57 (2007) no. 7, pp. 2183-2195.

It is well-known that the N-th Riemann sum of a compactly supported function on the real line converges to the Riemann integral at a much faster rate than the standard O(1/N) rate of convergence if the sum is over the lattice, Z/N. In this paper we prove an n-dimensional version of this result for Riemann sums over polytopes.

Il est bien connu que l’intégrale de Riemann d’une fonction d’une variable est beaucoup mieux approximée par la N-ième somme de Riemann si la somme est effectuée sur le réseau Z/N. Dans cet article nous démontrons un résultat similaire en plusieurs variables pour des sommes de Riemann sur des polytopes.

DOI: 10.5802/aif.2330
Classification: 52B20
Keywords: Riemann sums, Euler-Maclaurin formula for polytopes, Ehrhart’s theorem
Mot clés : sommes de Riemann, formule d’Euler-Maclaurin pour les polytopes, théorème de Ehrhart

Guillemin, Victor 1; Sternberg, Shlomo 2

1 MIT Department of Mathematics Cambridge, MA 02139 (USA)
2 Harvard University Department of Mathematics Cambridge, MA 02140 (USA)
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Guillemin, Victor; Sternberg, Shlomo. Riemann sums over polytopes. Annales de l'Institut Fourier, Volume 57 (2007) no. 7, pp. 2183-2195. doi : 10.5802/aif.2330. https://aif.centre-mersenne.org/articles/10.5802/aif.2330/

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