Riemann sums over polytopes
Annales de l'Institut Fourier, Volume 57 (2007) no. 7, p. 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 : https://doi.org/10.5802/aif.2330
Classification:  52B20
Keywords: Riemann sums, Euler-Maclaurin formula for polytopes, Ehrhart’s theorem
@article{AIF_2007__57_7_2183_0,
     author = {Guillemin, Victor and Sternberg, Shlomo},
     title = {Riemann sums over polytopes},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {57},
     number = {7},
     year = {2007},
     pages = {2183-2195},
     doi = {10.5802/aif.2330},
     mrnumber = {2394539},
     zbl = {1143.52011},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2007__57_7_2183_0}
}
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. http://www.numdam.org/item/AIF_2007__57_7_2183_0/

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