Borel's lemma

From formulasearchengine
Jump to navigation Jump to search

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

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 an 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

Proofs of Borel's lemma can be found in many text books on analysis, including Template:Harvtxt and Template:Harvtxt, 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 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.

See also

References

  • {{#invoke:citation/CS1|citation

|CitationClass=citation }}

  • {{#invoke:citation/CS1|citation

|CitationClass=citation }}

  • {{#invoke:citation/CS1|citation

|CitationClass=citation }}

This article incorporates material from Borel lemma on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.