It is well-known that the -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 rate of convergence if the sum is over the lattice, . 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 -ième somme de Riemann si la somme est effectuée sur le réseau . Dans cet article nous démontrons un résultat similaire en plusieurs variables pour des sommes de Riemann sur des polytopes.
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
@article{AIF_2007__57_7_2183_0, author = {Guillemin, Victor and Sternberg, Shlomo}, title = {Riemann sums over polytopes}, journal = {Annales de l'Institut Fourier}, pages = {2183--2195}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {57}, number = {7}, year = {2007}, doi = {10.5802/aif.2330}, zbl = {1143.52011}, mrnumber = {2394539}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.2330/} }
TY - JOUR AU - Guillemin, Victor AU - Sternberg, Shlomo TI - Riemann sums over polytopes JO - Annales de l'Institut Fourier PY - 2007 SP - 2183 EP - 2195 VL - 57 IS - 7 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2330/ DO - 10.5802/aif.2330 LA - en ID - AIF_2007__57_7_2183_0 ER -
%0 Journal Article %A Guillemin, Victor %A Sternberg, Shlomo %T Riemann sums over polytopes %J Annales de l'Institut Fourier %D 2007 %P 2183-2195 %V 57 %N 7 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2330/ %R 10.5802/aif.2330 %G en %F 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://archive.numdam.org/articles/10.5802/aif.2330/
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