Borel's lemma

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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 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

\displaystyle{\left(\frac{\partial^k}{\partial t^k}F\right)(0,x) = f_k(x),}

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 δ⋅Zn, it can be assumed that all the fm have compact support in some fixed closed ball C. For each m, let

\displaystyle{F_m(t,x)={t^m\over m!} \cdot \psi\left({t\over \varepsilon_m}\right)\cdot f_m(x),}

where εm is chosen sufficiently small that

\displaystyle{\|\partial^\alpha F_m \|_\infty \le 2^{-m}}

for |α| < m. These estimates imply that each sum

\displaystyle{\sum_{m\ge 0}  \partial^\alpha F_m}

is uniformly convergent and hence that

\displaystyle{F=\sum_{m\ge 0} F_m}

is a smooth function with

\displaystyle{\partial^\alpha F=\sum_{m\ge 0} \partial^\alpha F_m.}

By construction

\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[edit]

References[edit]

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