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 an 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

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

for k ≥ 0 and x in U.

See also

• non-analytic smooth functions.

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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