Let $f_n : U \rightarrow \mathbb R$ be a given sequence of functions of class $C^\infty$ on open subsets $U \subset \mathbb R^n$. Does there exist a function $F:\mathbb R \times U \rightarrow \mathbb R$ of class $C^\infty$ such that $$ \frac{\partial^n f}{\partial t^n}(0, x)=f_n(x) $$ for $n=0,1,2,\ldots $ and all $x \in U ?$

1$\begingroup$ en.wikipedia.org/wiki/Borel%27s_lemma $\endgroup$ – Willie Wong Oct 8 '12 at 10:33

$\begingroup$ This looks to me like a parametrised version of Borel's theorem. Am I interpreting it correctly? $\endgroup$ – Loop Space Oct 8 '12 at 10:34

$\begingroup$ Thanks for help. I did not know earlier that it is Borel's lemma. $\endgroup$ – user 12345 Oct 8 '12 at 10:49
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This is Theorem 1.2.6 in L. Hormander, Analysis of linear partial differential operators, vol. I.

$\begingroup$ Yes, but in Hormander books is assumed additionally that each $f_n$ has compact support. $\endgroup$ – user 12345 Oct 8 '12 at 19:14

$\begingroup$ This is not important. Using the standard technique, partition of unity, (Hormander, section I.1.4) one gets rid of the assumption of compact support. $\endgroup$ – Alexandre Eremenko Oct 9 '12 at 14:20