Note that it suffices to prove the result for a small interval I = (−ε,ε), since if ψ(t) is a smooth bump function with compact support in (−ε,ε) equal identically to 1 near 0, then ψ(t) ⋅ F(t, x) gives a solution on R × U. Similarly using a smooth partition of unity on Rn subordinate to a covering by open balls with centres at δ⋅Zn, it can be assumed that all the fm have compact support in some fixed closed ball C. For each m, let
where εm is chosen sufficiently small that
for |α| < m. These estimates imply that each sum
is uniformly convergent and hence that
is a smooth function with
By construction
Note: Exactly the same construction can be applied, without the auxiliary space U, to produce a smooth function on the interval I for which the derivatives at 0 form an arbitrary sequence.
Hörmander, Lars (1990), The analysis of linear partial differential operators, I. Distribution theory and Fourier analysis (2nd ed.), Springer-Verlag, p. 16, ISBN3-540-52343-X
borel, lemma, mathematics, named, after, Émile, borel, important, result, used, theory, asymptotic, expansions, partial, differential, equations, contents, statement, proof, also, referencesstatement, editsuppose, open, euclidean, space, suppose, that, sequenc. In mathematics Borel s lemma named after Emile Borel is an important result used in the theory of asymptotic expansions and partial differential equations Contents 1 Statement 2 Proof 3 See also 4 ReferencesStatement EditSuppose U is an open set in the Euclidean space Rn and suppose that f0 f1 is a sequence of smooth functions on U If I is any open interval in R containing 0 possibly I R then there exists a smooth function F t x defined on I U such that k F t k 0 x f k x displaystyle left frac partial k F partial t k right 0 x f k x for k 0 and x in U Proof EditProofs of Borel s lemma can be found in many text books on analysis including Golubitsky amp Guillemin 1974 and Hormander 1990 from which the proof below is taken Note that it suffices to prove the result for a small interval I e e since if ps t is a smooth bump function with compact support in e e equal identically to 1 near 0 then ps t F t x gives a solution on R U Similarly using a smooth partition of unity on Rn subordinate to a covering by open balls with centres at d Zn it can be assumed that all the fm have compact support in some fixed closed ball C For each m let F m t x t m m ps t e m f m x displaystyle F m t x t m over m cdot psi left t over varepsilon m right cdot f m x where em is chosen sufficiently small that a F m 2 m displaystyle partial alpha F m infty leq 2 m for a lt m These estimates imply that each sum m 0 a F m displaystyle sum m geq 0 partial alpha F m is uniformly convergent and hence that F m 0 F m displaystyle F sum m geq 0 F m is a smooth function with a F m 0 a F m displaystyle partial alpha F sum m geq 0 partial alpha F m By construction t m F t x t 0 f m x displaystyle partial t m F t x t 0 f m x Note Exactly the same construction can be applied without the auxiliary space U to produce a smooth function on the interval I for which the derivatives at 0 form an arbitrary sequence See also EditNon analytic smooth function Application to Taylor seriesReferences EditErdelyi A 1956 Asymptotic expansions Dover Publications pp 22 25 ISBN 0486603180 Golubitsky M Guillemin V 1974 Stable mappings and their singularities Graduate Texts in Mathematics vol 14 Springer Verlag ISBN 0 387 90072 1 Hormander Lars 1990 The analysis of linear partial differential operators I Distribution theory and Fourier analysis 2nd ed Springer Verlag p 16 ISBN 3 540 52343 XThis article incorporates material from Borel lemma on PlanetMath which is licensed under the Creative Commons Attribution Share Alike License Retrieved from https en wikipedia org w index php title Borel 27s lemma amp oldid 1037606018, wikipedia, wiki, book, books, library,
