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Does there exist a smooth bump function $\Phi: \mathbb{R} \rightarrow \mathbb{R} $ such that the sequence $M_k = \max_{x \in \mathbb{R}}|\Phi^{(k)}(x)|$ does not grow faster than exponentially? In general, are there examples of $\Phi$ where the $M_k$ grow (much) slower than in the case of $e^{\frac{1}{x^2-1}}$ (or the like)?

By smooth bump function, I mean a $C^\infty$ function that is $1$ in a neighbourhood of $0$ with compact support.

Pointers to other resources on this are appreciated as well.

guest123
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  • Please take a look on that answer by Robert Israel: https://math.stackexchange.com/questions/101480/are-there-other-kinds-of-bump-functions-than-e-frac1x2-1#101484 – Hasek Jul 22 '17 at 21:18
  • I don't understand why everyone is marking this as a duplicate. I'm not just asking for other bump functions, I'm asking for a bump function who's derivatives don't grow too fast, as made precisely in the original question statement. – guest123 Jul 28 '17 at 13:21
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    @guest123 Almost 6y late, but try to build your bump function as in the standard case but consider $f(x)\times e^{\frac{1}{1-x^2}}$ instead of $ e^{\frac{1}{1-x^2}}$. Play with $f$ to controll the growth / the derivative – Didier Jan 05 '22 at 12:41
  • A bit late too, but the answer to this question shows that the function you are looking for doesn´t exist – Saúl RM Jan 05 '22 at 17:41

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