The Discrete Analog of the Malgrange-Ehrenpreis Theorem

By Doron Zeilberger

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[Appeared in: "From Fourier Analysis and Number Theory to Radon Transforms and Geometry-In Memory of Leon Ehrenpreis" (Developments in Mathematics) edited by M. Farkas, R. C. Gunning, Marvin I. Knopp z"l, and B.A. Taylor, Springer, published Sept. 2012]
In fond memory of Leon Ehrenpreis (1930-2010)
Written: July 21, 2011
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.

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