Algebraic Birkhoff Factorization and the Euler-Maclaurin Formula on Cones

TL;DR

This paper develops an algebraic framework for lattice cones, using Birkhoff factorization to derive a generalized Euler-Maclaurin formula that connects exponential sums and integrals.

Contribution

It introduces a coproduct structure on lattice cones and applies algebraic Birkhoff factorization to relate exponential sums and integrals via subdivision properties.

Findings

Factorization of exponential sum as a convolution quotient

Derivation of a generalized Euler-Maclaurin formula on cones

A simple formula for the interpolating factor using projection

Abstract

We equip the space of lattice cones with a coproduct which makes it a connected cograded colagebra. The exponential sum and exponential integral on lattice cones can be viewed as linear maps on this space with values in the space of meromorphic germs with linear poles at zero. We investigate the subdivision properties-- reminiscent of the inclusion-exclusion principle for the cardinal on finite sets-- of such linear maps and establish a compatibility of these properties with respect to the convolution quotient of the coalgebra. Implementing the Algebraic Birkhoff Factorization procedure on the linear maps under consideration, we factorize the exponential sum as a convolution quotient of two maps, with each of the maps in the factorization satisfying a subdivision property. Consequently, the Algebraic Birkhoff Factorization specializes to the Euler-Maclaurin formula on lattice cones and provides a simple formula for the interpolating factor by means of a projection map.