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 R<sup>n</sup>, and suppose that f<sub>0</sub>, f<sub>1</sub>, ... 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
:<math>\left.\frac{\partial^k F}{\partial t^k}\right|_{(0,x)} = f_k(x),</math>
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 R<sup>n</sup> subordinate to a covering by open balls with centres at δ⋅Z<sup>n</sup>, it can be assumed that all the f<sub>m</sub> have compact support in some fixed closed ball C. For each m, let
:<math>F_m(t,x)={t^m\over m!} \cdot \psi\left({t\over \varepsilon_m}\right)\cdot f_m(x),</math>
where ε<sub>m</sub> is chosen sufficiently small that
:<math>\|\partial^\alpha F_m \|_\infty \le 2^{-m}</math>
for |α| < m. These estimates imply that each sum
:<math>\sum_{m\ge 0} \partial^\alpha F_m</math>
is uniformly convergent and hence that
:<math>F=\sum_{m\ge 0} F_m</math>
is a smooth function with
:<math>\partial^\alpha F=\sum_{m\ge 0} \partial^\alpha F_m.</math>
By construction
:<math>\partial_t^m F(t,x)|_{t=0}=f_m(x).</math>
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.