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To be specific, suppose we are given a sequence of smooth functions $\{f_k\}_{k\geq 0}$ on flat torus $\mathbf{T}^2$(you may think of it as doubly periodic functions on $\mathbf{R}^2$ and smooth).

Question. How can we construct a smooth function $f(x,t)$ on $\mathbf{T}^2\times[0,1]$ such that $\frac{\partial^k}{\partial t^k}|_{t=0}f(x,t)=f_k(x)$ for any $k\geq0$? Is this always possible?

Of course, if one just prescribes finitely many derivatives or the sequence $f_k$ satisfies a bound in $k$ which makes $f(t,x)$ analytic in $t$, I know how to do it. But I don`t know if this can be done in general.

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Yes, this is known as Borel's Lemma (it is often stated only for open sets $U\subseteq\mathbb R^n$, but you can reduce your case of $T^2$ to this one using a partition of unity argument).

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  • $\begingroup$ Thanks for your answer! I have just thought of this. Actually the construction works for my case and no partition of unity is needed. $\endgroup$ – Jingrui Cheng Jun 3 '15 at 18:00

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