TY - JOUR

T1 - Asymptotic Euler-Maclaurin formula over lattice polytopes

AU - Tate, Tatsuya

N1 - Funding Information:
✩ Research partially supported by JSPS Grant-in-Aid for Scientific Research (No. 21740117). E-mail address: tate@math.nagoya-u.ac.jp.

PY - 2011/1/15

Y1 - 2011/1/15

N2 - Formulas for the Riemann sums over lattice polytopes determined by the lattice points in the polytopes are often called Euler-Maclaurin formulas. An asymptotic Euler-Maclaurin formula, by which we mean an asymptotic expansion formula for Riemann sums over lattice polytopes, was first obtained by Guillemin and Sternberg (2007) [11]. Then, the problem is to find a concrete formula for each term of the expansion. In this paper, an asymptotic Euler-Maclaurin formula of the Riemann sums over general lattice polytopes is given. The formula given here is an asymptotic form of the so-called local Euler-Maclaurin formula of Berline and Vergne (2007) [3]. For Delzant polytopes, our proof given here is independent of the local Euler-Maclaurin formula. Furthermore, a concrete description of differential operators which appear in each term of the asymptotic expansion for Delzant lattice polytopes is given. By using this description, when the polytopes are Delzant lattice, a concrete formula for each term of the expansion in two dimension and a formula for the third term of the expansion in arbitrary dimension are given.

AB - Formulas for the Riemann sums over lattice polytopes determined by the lattice points in the polytopes are often called Euler-Maclaurin formulas. An asymptotic Euler-Maclaurin formula, by which we mean an asymptotic expansion formula for Riemann sums over lattice polytopes, was first obtained by Guillemin and Sternberg (2007) [11]. Then, the problem is to find a concrete formula for each term of the expansion. In this paper, an asymptotic Euler-Maclaurin formula of the Riemann sums over general lattice polytopes is given. The formula given here is an asymptotic form of the so-called local Euler-Maclaurin formula of Berline and Vergne (2007) [3]. For Delzant polytopes, our proof given here is independent of the local Euler-Maclaurin formula. Furthermore, a concrete description of differential operators which appear in each term of the asymptotic expansion for Delzant lattice polytopes is given. By using this description, when the polytopes are Delzant lattice, a concrete formula for each term of the expansion in two dimension and a formula for the third term of the expansion in arbitrary dimension are given.

KW - Asymptotic expansion

KW - Euler-Maclaurin formula

KW - Lattice polytopes

KW - Toric varieties

UR - http://www.scopus.com/inward/record.url?scp=78049445159&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78049445159&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2010.08.011

DO - 10.1016/j.jfa.2010.08.011

M3 - Article

AN - SCOPUS:78049445159

VL - 260

SP - 501

EP - 540

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 2

ER -