Borel's lemma

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 δ⋅Zⁿ, it can be assumed that all the f_m 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 ε_m 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.

See also

• Non-analytic smooth function § Application to Taylor series

References

• Erdélyi, A. (1956), Asymptotic expansions, Dover Publications, pp. 22–25, ISBN 0486603180
• Golubitsky, M.; Guillemin, V. (1974), Stable mappings and their singularities, Graduate Texts in Mathematics, vol. 14, Springer-Verlag, ISBN 0-387-90072-1
• Hörmander, Lars (1990), The analysis of linear partial differential operators, I. Distribution theory and Fourier analysis (2nd ed.), Springer-Verlag, p. 16, ISBN 3-540-52343-X

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