The Discrete Analog of the Malgrange-Ehrenpreis Theorem

TL;DR

This paper establishes a discrete analog of the Malgrange-Ehrenpreis theorem, proving that every non-zero linear difference operator with constant coefficients has a fundamental solution, using duality and Laurent series methods.

Contribution

It introduces the discrete analog of the Malgrange-Ehrenpreis theorem and provides two proofs, one using duality and the other constructive with Laurent series.

Findings

Proves the existence of fundamental solutions for discrete linear difference operators.

Provides two distinct proofs of the discrete analog theorem.

Includes a Maple package for computational implementation.

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 .