[Paper Review] EULER-MACLAURIN FORMULAS VIA DIFFERENTIAL OPERATORS

Yohann Le Floch, Álvaro Pelayo|arXiv (Cornell University)|Dec 19, 2013

Mathematical functions and polynomials21 references4 citations

TL;DR

This paper derives explicit Euler-MacLaurin asymptotic expansions for Riemann sums over polytopes in dimensions one to three using differential operators, recovering and simplifying results from Tate (2010) via elementary methods rooted in Guillemin and Sternberg's work. The key contribution is expressing coefficients in the expansion as sums of normal-derivative differential operators on faces, valid for wedges, intervals, polygons, and 3D polytopes.

ABSTRACT

Recently there has been a renewed interest in asymptotic Euler-MacLaurin formulas, partly due to applications to spectral theory of differential operators. Using elementary means, we recover such formulas for compactly supported smooth functions f on intervals, polygons, and 3-dimensional polytopes, where the coefficients in the asymptotic expansion are sums of differential operators involving only derivatives of f in directions normal to the faces of the polytope. Our formulas apply to wedges of any dimension. This paper builds on, and is motivated by, works of Guillemin, Sternberg, and others, in the past ten years.

Motivation & Objective

• To recover and simplify Tate’s explicit Euler-MacLaurin formulas for Riemann sums over lattice polytopes using elementary methods.
• To express the coefficients in the asymptotic expansion as differential operators involving only normal derivatives of the function on the faces of the polytope.
• To extend the applicability of Euler-MacLaurin formulas to wedges and polytopes in dimensions one to three, with a focus on geometric and spectral applications.
• To provide a unified, computationally tractable framework for asymptotic approximations of sums over lattice points in polytopes, particularly relevant in spectral theory and geometric quantization.
• To establish a direct correspondence between Tate’s complex combinatorial formula and a cleaner, operator-based formulation using differential operators and face integrals.

Proposed method

• Utilizes an elementary derivation based on Guillemin and Sternberg’s earlier work on asymptotic expansions for lattice point sums.
• Applies differential operators constructed from Bernoulli numbers and normal derivatives along face normals to capture correction terms in the Euler-MacLaurin expansion.
• Expresses the asymptotic expansion of Riemann sums over polytopes as a sum over faces, with each term involving an integral of a differential operator applied to the function.
• Introduces a notation system that maps Tate’s combinatorial indexing of face contributions to a geometric index set based on active face indices.
• Uses the star-integral notation ∫* to denote integration over the intersection of half-spaces dual to non-zero face directions, capturing the local geometry of the polytope.
• Derives the expansion for wedges first, then extends it to general polytopes via decomposition and face-wise contributions, maintaining consistency with spectral theory applications.

Experimental results

Research questions

• RQ1 Can Euler-MacLaurin formulas for Riemann sums over polytopes be derived using only elementary differential operators, without heavy combinatorial machinery?
• RQ2 How can the coefficients in the asymptotic expansion be expressed purely in terms of normal derivatives of the function on the faces of the polytope?
• RQ3 What is the precise relationship between Tate’s combinatorial formula involving Bernoulli numbers and the differential operator formulation presented here?
• RQ4 To what extent can this method be generalized to higher-dimensional polytopes, and what are the computational barriers in dimensions n ≥ 4?
• RQ5 How does this operator-based approach simplify the computation of spectral sums in geometric quantization and toric manifold theory?

Key findings

• The paper derives a new, explicit Euler-MacLaurin formula for Riemann sums over wedges in ℝⁿ, where the coefficients are expressed as sums of differential operators involving only normal derivatives of the function on the faces.
• For intervals, polygons, and 3D polytopes, the asymptotic expansion is shown to be equivalent to Tate’s formula but reformulated using differential operators and face-based integrals.
• The key formula for wedges is given as R_N(W,f) ∼ ∑_{q≥0} N^{-q} ∑_{α∈ℕⁿ, |α|=q} (1/α!) (∏_{i=1}^n b_{α_i}) ∫*_{∩_{i:α_i>0} ℋ_i} D^{q−ν(α)}f · v_{α−r(α)}, where the integral is over the intersection of half-spaces dual to non-zero face directions.
• The method recovers Tate’s result for regular wedges in a more geometric and operator-theoretic form, with a direct correspondence established between Tate’s combinatorial indexing and the new differential operator formulation.
• The approach is shown to be extendable to higher dimensions, though the complexity of computations increases significantly beyond dimension 3.
• The framework provides a clean, systematic way to compute spectral sums in geometric quantization, particularly in the context of toric manifolds and Berezin-Toeplitz operators.