BREAKING NEWS
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

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

${\displaystyle \left.{\frac {\partial ^{k}F}{\partial t^{k}}}\right|_{(0,x)}=f_{k}(x),}$

for k ≥ 0 and x in U.

Proof

Proofs of Borel's lemma can be found in many text books on analysis, including 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

${\displaystyle F_{m}(t,x)={t^{m} \over m!}\cdot \psi \left({t \over \varepsilon _{m}}\right)\cdot f_{m}(x),}$

where εm is chosen sufficiently small that

${\displaystyle \|\partial ^{\alpha }F_{m}\|_{\infty }\leq 2^{-m}}$

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

${\displaystyle \sum _{m\geq 0}\partial ^{\alpha }F_{m}}$

is uniformly convergent and hence that

${\displaystyle F=\sum _{m\geq 0}F_{m}}$

is a smooth function with

${\displaystyle \partial ^{\alpha }F=\sum _{m\geq 0}\partial ^{\alpha }F_{m}.}$

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.