Malgrange Ehrenpreis Theorem
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, the Malgrange Ehrenpreis theorem states that every non-zero 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). The original proofs of Malgrange and Ehrenpreis were non-constructive as they used the Hahn Banach theorem. Since then several constructive proof have been found. There is a very short proof using the Fourier transform an...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, the Malgrange Ehrenpreis theorem states that every non-zero 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). The original proofs of Malgrange and Ehrenpreis were non-constructive as they used the Hahn Banach theorem. Since then several constructive proof have been found. There is a very short proof using the Fourier transform and the Bernstein Sato polynomial, as follows. By taking Fourier transforms the Malgrange Ehrenpreis theorem is equivalent to the fact that every non-zero polynomial P has a distributional inverse. By replacing P by the product with its complex conjugate, one can also assume that P is non-negative. For non-negative polynomials P the existence of a distributional inverse follows from the existence of the Bernstein Sato polynomial, which implies that Ps can be analytically continued as a meromorphic distribution-valued function of the complex variable s; the constant term of the Laurent expansion of Ps at s = 1 is then a distributional inverse of P.
