1
$\begingroup$

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 $x \in U ?$

$\endgroup$
  • 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
1
$\begingroup$

This is Theorem 1.2.6 in L. Hormander, Analysis of linear partial differential operators, vol. I.

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.