TY - JOUR
T1 - Exact Euler-Maclaurin formulas for simple lattice polytopes
AU - Karshon, Yael
AU - Sternberg, Shlomo
AU - Weitsman, Jonathan
N1 - Funding Information:
✩ This work was partially supported by United States–Israel Binational Science Foundation Grant number 2000352 (to Y.K. and J.W.), by the Connaught Fund (to Y.K.), and by National Science Foundation Grant DMS 99/71914 and DMS 04/05670 (to J.W.). * Corresponding author. E-mail addresses: [email protected] (Y. Karshon), [email protected] (S. Sternberg), [email protected] (J. Weitsman).
PY - 2007/7
Y1 - 2007/7
N2 - Euler-Maclaurin formulas for a polytope express the sum of the values of a function over the lattice points in the polytope in terms of integrals of the function and its derivatives over faces of the polytope or its expansions. Exact Euler-Maclaurin formulas [A.G. Khovanskii, A.V. Pukhlikov, Algebra and Analysis 4 (1992) 188-216; S.E. Cappell, J.L. Shaneson, Bull. Amer. Math. Soc. 30 (1994) 62-69; C. R. Acad. Sci. Paris Sér. I Math. 321 (1995) 885-890; V. Guillemin, J. Differential Geom. 45 (1997) 53-73; M. Brion, M. Vergne, J. Amer. Math. Soc. 10 (2) (1997) 371-392] apply to exponential or polynomial functions; Euler-Maclaurin formulas with remainder [Y. Karshon, S. Sternberg, J. Weitsman, Proc. Natl. Acad. Sci. 100 (2) (2003) 426-433; Duke Math. J. 130 (3) (2005) 401-434] apply to more general smooth functions. In this paper we review these results and present proofs of the exact formulas obtained by these authors, using elementary methods. We then use an algebraic formalism due to Cappell and Shaneson to relate the different formulas.
AB - Euler-Maclaurin formulas for a polytope express the sum of the values of a function over the lattice points in the polytope in terms of integrals of the function and its derivatives over faces of the polytope or its expansions. Exact Euler-Maclaurin formulas [A.G. Khovanskii, A.V. Pukhlikov, Algebra and Analysis 4 (1992) 188-216; S.E. Cappell, J.L. Shaneson, Bull. Amer. Math. Soc. 30 (1994) 62-69; C. R. Acad. Sci. Paris Sér. I Math. 321 (1995) 885-890; V. Guillemin, J. Differential Geom. 45 (1997) 53-73; M. Brion, M. Vergne, J. Amer. Math. Soc. 10 (2) (1997) 371-392] apply to exponential or polynomial functions; Euler-Maclaurin formulas with remainder [Y. Karshon, S. Sternberg, J. Weitsman, Proc. Natl. Acad. Sci. 100 (2) (2003) 426-433; Duke Math. J. 130 (3) (2005) 401-434] apply to more general smooth functions. In this paper we review these results and present proofs of the exact formulas obtained by these authors, using elementary methods. We then use an algebraic formalism due to Cappell and Shaneson to relate the different formulas.
UR - http://www.scopus.com/inward/record.url?scp=34247471252&partnerID=8YFLogxK
U2 - 10.1016/j.aam.2006.04.003
DO - 10.1016/j.aam.2006.04.003
M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???
AN - SCOPUS:34247471252
SN - 0196-8858
VL - 39
SP - 1
EP - 50
JO - Advances in Applied Mathematics
JF - Advances in Applied Mathematics
IS - 1
ER -