Abstract
One of the landmarks of the modern theory of partial differential equations is the Malgrange–Ehrenpreis theorem that states that every nonzero 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.
Mathematics Subject Classification(2010): 35E05 (Primary), 39A06 (Secondary)
First version: July 21, 2011. This version: Sept. 7, 2011.
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References
Leon Ehrenpreis, Solution of some problems of division. I. Division by a polynomial of derivation, Amer. J. Math. 76(1954), 883–903.
Leon Ehrenpreis, Solution of some problems of division. II. Division by a punctual distribution, Amer. J. Math. 77(1955), 286–292.
Bernard Malgrange, Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, Ann. Inst. Fourier, Grenoble 6(1955-1956): 271–355.
Peter Wagner, A new constructive proof of the Malgrange–Ehrenpreis theorem, Amer. Math. Monthly 116(2009), 457–462.
Wikipedia,the free Encyclopedia, Malgrange–Ehrenpreis Theorem, Retrieved 16:10, July 21, 2011.
Acknowledgements
I’d like to thank an anonymous referee, and Hershel Farkas, for insightful comments. Accompanied by Maple package LEON available from http://www.math.rutgers.edu/\~zeilberg/tokhniot/LEON. Supported in part by the USA National Science Foundation.
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In fond memory of Leon Ehrenpreis
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Zeilberger, D. (2013). The Discrete Analog of the Malgrange–Ehrenpreis Theorem. In: Farkas, H., Gunning, R., Knopp, M., Taylor, B. (eds) From Fourier Analysis and Number Theory to Radon Transforms and Geometry. Developments in Mathematics, vol 28. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4075-8_27
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DOI: https://doi.org/10.1007/978-1-4614-4075-8_27
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