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

