Borel's lemma
Appearance
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
for k ≥ 0 and x in U.
See also
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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