|
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 and . This means that the differential equation : where ''P'' is a polynomial in several variables and δ is the Dirac delta function, has a distributional solution ''u''. It can be used to show that : has a solution for any distribution ''f''. 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. ==Proofs== The original proofs of Malgrange and Ehrenpreis were non-constructive as they used the Hahn–Banach theorem. Since then several constructive proofs 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 ''P''''s'' can be analytically continued as a meromorphic distribution-valued function of the complex variable ''s''; the constant term of the Laurent expansion of ''P''''s'' at ''s'' = −1 is then a distributional inverse of ''P''. Other proofs, often giving better bounds on the growth of a solution, are given in , and . gives a detailed discussion of the regularity properties of the fundamental solutions. A short constructive proof was presented in : : is a fundamental solution of ''P''(∂), i.e., ''P''(∂)''E'' = δ, if ''Pm'' is the principal part of ''P'', η ∈ R''n'' with ''Pm''(η) ≠ 0, the real numbers λ0, ..., λ''m'' are pairwise different, and : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Malgrange–Ehrenpreis theorem」の詳細全文を読む スポンサード リンク
|