Malgrange–Ehrenpreis theorem
E1021737
The Malgrange–Ehrenpreis theorem is a fundamental result in the theory of partial differential equations stating that every linear partial differential operator with constant coefficients admits a fundamental solution.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Malgrange–Ehrenpreis theorem canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T13070902 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Malgrange–Ehrenpreis theorem Context triple: [Bernard Malgrange, notableFor, Malgrange–Ehrenpreis theorem]
-
A.
Malgrange preparation theorem
The Malgrange preparation theorem is a fundamental result in analysis and singularity theory that generalizes the Weierstrass preparation theorem to smooth functions, providing a local factorization of such functions near singular points.
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B.
Mittag-Leffler theorem
The Mittag-Leffler theorem is a fundamental result in complex analysis that characterizes meromorphic functions by allowing the construction of such functions with prescribed principal parts at given poles.
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C.
Weierstrass preparation theorem
The Weierstrass preparation theorem is a fundamental result in complex analysis and analytic geometry that locally expresses analytic functions near a zero as a product of a polynomial and a unit, enabling a power-series analogue of factorization.
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D.
Picard theorem
Picard theorem is a fundamental result in complex analysis stating that entire non-constant functions take on all possible complex values, with at most one exception.
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E.
Cauchy–Kovalevskaya theorem
The Cauchy–Kovalevskaya theorem is a fundamental result in partial differential equations that guarantees the existence and uniqueness of analytic solutions to certain initial value problems under appropriate analyticity conditions.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Malgrange–Ehrenpreis theorem Target entity description: The Malgrange–Ehrenpreis theorem is a fundamental result in the theory of partial differential equations stating that every linear partial differential operator with constant coefficients admits a fundamental solution.
-
A.
Malgrange preparation theorem
The Malgrange preparation theorem is a fundamental result in analysis and singularity theory that generalizes the Weierstrass preparation theorem to smooth functions, providing a local factorization of such functions near singular points.
-
B.
Mittag-Leffler theorem
The Mittag-Leffler theorem is a fundamental result in complex analysis that characterizes meromorphic functions by allowing the construction of such functions with prescribed principal parts at given poles.
-
C.
Weierstrass preparation theorem
The Weierstrass preparation theorem is a fundamental result in complex analysis and analytic geometry that locally expresses analytic functions near a zero as a product of a polynomial and a unit, enabling a power-series analogue of factorization.
-
D.
Picard theorem
Picard theorem is a fundamental result in complex analysis stating that entire non-constant functions take on all possible complex values, with at most one exception.
-
E.
Cauchy–Kovalevskaya theorem
The Cauchy–Kovalevskaya theorem is a fundamental result in partial differential equations that guarantees the existence and uniqueness of analytic solutions to certain initial value problems under appropriate analyticity conditions.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem in partial differential equations ⓘ |
| appliesTo |
operators on Euclidean space R^n
ⓘ
systems of linear PDEs with constant coefficients ⓘ |
| area |
analysis
ⓘ
theory of linear operators ⓘ |
| assumes | coefficients of the differential operator are constant ⓘ |
| classification | existence theorem ⓘ |
| concerns |
fundamental solutions of partial differential operators
ⓘ
linear partial differential operators with constant coefficients ⓘ |
| context | modern theory of linear PDEs ⓘ |
| doesNotRequire | ellipticity of the operator ⓘ |
| field |
distribution theory
ⓘ
functional analysis ⓘ partial differential equations ⓘ |
| generalizes | existence of Green’s functions for ordinary differential equations with constant coefficients ⓘ |
| guarantees | existence of fundamental solutions for all constant coefficient linear PDEs ⓘ |
| hasConsequence | every constant coefficient linear PDE is locally solvable in the sense of distributions ⓘ |
| historicalPeriod | 20th century mathematics ⓘ |
| implies | existence of a distribution E such that P(D)E = δ for any constant coefficient operator P(D) ⓘ |
| importance | provides a general method to construct solutions to linear PDEs via convolution with a fundamental solution ⓘ |
| influenced |
development of distribution theory in PDE
ⓘ
subsequent work on variable coefficient operators ⓘ |
| involves |
Dirac delta distribution
ⓘ
Fourier–Laplace transform ⓘ polynomials in several variables ⓘ |
| isFundamentalResultIn | theory of linear partial differential equations with constant coefficients ⓘ |
| language | distribution theory ⓘ |
| namedAfter |
Bernard Malgrange
ⓘ
Leon Ehrenpreis ⓘ |
| proofTechnique |
Fourier analytic methods
ⓘ
complex analysis in several variables ⓘ |
| provedBy |
Bernard Malgrange
ⓘ
Leon Ehrenpreis ⓘ |
| relatedResult |
Lax–Malgrange theorem
ⓘ
Paley–Wiener–Schwartz theorem ⓘ
linked to:
Paley–Wiener theorem
|
| relatesTo |
Cauchy problem for linear partial differential equations
ⓘ
fundamental solution method ⓘ solvability of linear PDEs ⓘ |
| statement | Every linear partial differential operator with constant coefficients admits a fundamental solution in the sense of distributions. ⓘ |
| typicalFormulation | For every nonzero polynomial P in n variables, there exists a distribution E on R^n such that P(D)E = δ_0. ⓘ |
| usesConcept |
Fourier transform
ⓘ
Schwartz distributions ⓘ convolution ⓘ tempered distributions ⓘ |
| yearProvedApprox | 1954 ⓘ |
How these facts were elicited
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Input
Subject: Malgrange–Ehrenpreis theorem Description of subject: The Malgrange–Ehrenpreis theorem is a fundamental result in the theory of partial differential equations stating that every linear partial differential operator with constant coefficients admits a fundamental solution.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.