Borel's lemma explained

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 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

\left. (\partial ^{k} F)/(\partial t ^{k}) \right| _{(0,x)} = f _{k(x),}

for k ≥ 0 and x in U.

Proof

Proofs of Borel's lemma can be found in many text books on analysis, including and, 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 f_m have compact support in some fixed closed ball C. For each m, let

F ^{m\over} _{m(t,x) = {t} m!} ⋅ \psi\left({t\over \varepsilon _{m}\right) ⋅} f _{m(x),}

where ε_m is chosen sufficiently small that

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

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

\sum _{m\ge} \partial ^{\alpha} F _{m}

is uniformly convergent and hence that

F = \sum _{m\ge} F _{m}

is a smooth function with

\partial ^{\alpha} F = \sum _{m\ge} \partial ^{\alpha} F _{m.}

By construction

\partial ^{m} _{t} 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.