In mathematics, the Malgrange–Ehrenpreis theorem states that every non-zero linear differential operator with

constant coefficients In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b ...

has a

Green's function In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differenti ...

. It was first proved independently by and . This means that the

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

P\left(\frac, \ldots,  \frac \right) u(\mathbf) = \delta(\mathbf),

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 :

P\left(\frac, \ldots,  \frac \right) u(\mathbf) = f(\mathbf)

has a solution for any compactly supported 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 The Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear f ...

. Since then several constructive proofs have been found. There is a very short proof using the Fourier transform and the

Bernstein–Sato polynomial In mathematics, the Bernstein–Sato polynomial is a polynomial related to differential operators, introduced independently by and , . It is also known as the b-function, the b-polynomial, and the Bernstein polynomial, though it is not related ...

, 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 : :

E=\frac \sum_^m a_j e^ \mathcal^_\left(\frac\right)

is a fundamental solution of ''P''(∂), i.e., ''P''(∂)''E'' = δ, if ''P_m'' is the principal part of ''P'', ''η'' ∈ R^''n'' with ''P''_''m''(''η'') ≠ 0, the real numbers ''λ''₀, ..., ''λ''_''m'' are pairwise different, and :

a_j=\prod_^m(\lambda_j-\lambda_k)^.

References

* * * * * * * * * {{DEFAULTSORT:Malgrange-Ehrenpreis theorem Differential equations Theorems in analysis Schwartz distributions