Borel's lemma

Template:Short description 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

${\frac{{\partial }^{k}F}{\partial t^k}|}_{(0,x)}=f_k(x),$

for k ≥ 0 and x in U.

Proof

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

$F_m(t,x)=\frac{t^m}{m!}\cdot \psi \left(\frac{t}{{\epsilon }_m}\right)\cdot f_m(x),$

where ε_m is chosen sufficiently small that

$\Vert {\partial }^{\alpha }{F}_m{\Vert }_{\mathrm{\infty }}\le 2^{-m}$

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

$\sum _{m\ge 0}{\partial }^{\alpha }{F}_m$

is uniformly convergent and hence that

$F=\sum _{m\ge 0}{F}_m$

is a smooth function with

${\partial }^{\alpha }F=\sum _{m\ge 0}{\partial }^{\alpha }{F}_m.$

By construction

$\underset{t}{\overset{m}{\partial }}F(t,x){|}_{t=0}=f_m(x).$

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

• Template:Slink

References

• Template:Citation
• Template:Citation
• Template:Citation