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 $x \in U ?$
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1$\begingroup$ en.wikipedia.org/wiki/Borel%27s_lemma $\endgroup$– Willie WongOct 8, 2012 at 10:33
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$\begingroup$ This looks to me like a parametrised version of Borel's theorem. Am I interpreting it correctly? $\endgroup$– Andrew StaceyOct 8, 2012 at 10:34
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$\begingroup$ Thanks for help. I did not know earlier that it is Borel's lemma. $\endgroup$– user 12345Oct 8, 2012 at 10:49
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1 Answer
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This is Theorem 1.2.6 in L. Hormander, Analysis of linear partial differential operators, vol. I.
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$\begingroup$ Yes, but in Hormander books is assumed additionally that each $f_n$ has compact support. $\endgroup$ Oct 8, 2012 at 19:14
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$\begingroup$ 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. $\endgroup$ Oct 9, 2012 at 14:20