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)
@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/

[1] Brion, M.; Vergne, M. Lattice points in simple polytopes, Jour. Amer. Math. Soc., Volume 10 (1997), pp. 371-392 | DOI | MR | Zbl

[2] Cappell, S.; Shaneson, J. Euler-Maclaurin expansions for lattices above dimension one, C. R. Acad. Sci. Paris Ser. I Math., Volume 321 (1995), pp. 885-890 | MR | Zbl

[3] Danilov, V. I. The geometry of toric varieties, Russ. Math. Surv., Volume 33 (1978) no. 2, pp. 97-154 | DOI | MR | Zbl

[4] Guillemin, V. Riemann-Roch for toric orbifolds, J. Differential Geom., Volume 45 (1997), pp. 53-73 | MR | Zbl

[5] Guillemin, V.; Sternberg, Shlomo; Weitsman, Jonathan The Ehrhart function for symbols (to appear)

[6] Guillemin, V.; Stroock, D. W. Some Riemann sums are better than others (to appear)

[7] Kantor, J. M.; Khovanskii, A. G. Une application du théorème de Riemann-Roch combinatoire au polynôme d’Ehrhart des polytopes entiers de R d , C. R. Acad. Sci. Paris Ser. I Math, Volume 317 (1993) no. 5, pp. 501-507 | Zbl

[8] Karshon, Y.; Sternberg, S.; Weitsman, J. Euler-MacLaurin with remainder for a simple integral polytope, Duke Mathematical Journal, Volume 130 (2005), pp. 401-434 | DOI | MR | Zbl

[9] Khovanskii, A. G; Pukhlikov, A. V. The Riemann-Roch theorem for integrals and sums of quasipolynomials on virtual polytopes, Algebra and Analysis, Volume 4 (1992), pp. 188-216 translation in St. Petersburg Math. J. (1993), no. 4, 789–812 | MR | Zbl

Cited by Sources: