Malgrange–Ehrenpreis theorem
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In mathematics, the Malgrange–Ehrenpreis theorem states that every non-zero linear differential operator with constant coefficients has a Green's function. It was first proved independently by Leon Ehrenpreis (1954, 1955) and Bernard Malgrange (1955–1956).
This means that the differential equation
where is a polynomial in several variables and is the Dirac delta function, has a distributional solution . It can be used to show that
has a solution for any compactly supported distribution . The solution is not unique in general.
The analogue for differential operators whose coefficients are polynomials (rather than constants) is false: see Lewy's example.