BOREL'S LEMMA

In mathematics, 'Borel's lemma' is an important result about partial differential equations named after Émile Borel.
Suppose $U$ is an open set in the Euclidean space 'R'ⁿ, and suppose that $f\_0,\; f\_1,\; ...$ is a sequence of smooth, complex-valued functions on $U$. Then there exists a smooth function $F=F(t,x)$ defined on 'R'×$U$ with complex values, such that
:
ight)(0,x) = f_k(x),
for all $k=0,1,...$, and $x$ in $U.$
A constructive proof of this result is given in Golubitsky (1974).

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References

References

★ M. Golubitsky, V. Guillemin (1974). ''Stable mappings and their singularities''. Springer-Verlag, Graduate texts in Mathematics: Vol. 14. ISBN 0-387-90072-1.