The Discrete Analog of the Malgrange-Ehrenpreis Theorem

Abstract

One of the landmarks of the modern theory of partial differential equations is the Malgrange- Ehrenpreis theorem that states that every non-zero linear partial differential operator with constant coefficients has a Green function (alias fundamental solution). In this short note I state the discrete analog, and give two proofs. The first one is Ehrenpreis- style, using duality, and the second one is constructive, using formal Laurent series. This article is accompanied by the Maple package LEON available from: http://www.math.rutgers.edu/~zeilberg/tokhniot/LEON .