Borel's lemma

Summary

In mathematics, Borel's lemma, named after Émile Borel, is an important result used in the theory of asymptotic expansions and partial differential equations.

Statement edit

Suppose 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

 

for k ≥ 0 and x in U.

Proof edit

Proofs of Borel's lemma can be found in many text books on analysis, including Golubitsky & Guillemin (1974) and Hörmander (1990), from which the proof below is taken.

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 δZn, 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.

See also edit

References edit

  • Erdélyi, 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
  • Hörmander, 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-X

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