# Is this (classical?) exercice missing a hypothesis?

A friend just told me about an exercice he was given quite a few years ago, but he wasn't sure wether he remembered all the hypothesis correctly. Does anybody recognize this?

Let $f$ be a smooth real-valued function on $\Bbb R$ such that for all $n\in\Bbb N,~\sup_{\Bbb R}|f^{(n)}|=1$ and $f(0)=1$, then $f=\cos$.

He wasn't sure wether theose were the exact hypothesis needed for this to work. He noticed that $f$ is automatically a power series with infinite radius of convergence, but wondered wether one ought to impose $f$ to be even, and wether the condition on the suprermums of the derivatives is correct.

NOTE. If you know the hypothesis, please post them as an answer, and if you want to post a solution, please hide the text so that one needs to scroll over it to reveal it, thanks!

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Hum... if this is true, it is quite striking, +1. And I wonder in which spheres this is classical. Not mine. – julien Mar 4 at 20:56
@Julien I agree :D I have tried a bit, and tried to prove that $f''(0)=-1$ which would be enough, but I don't make any progress, and doubt I'm on the right path. – Olivier Bégassat Mar 4 at 21:17
I understand that if you can prove $f''(0)=-1$, then you apply this to $-f''$ to find $f^{(4)}(0)=1$ and so on. But you are probably also assuming $f$ even, in which case the result follows clearly. Otherwise, how do deduce it? – julien Mar 4 at 21:27
@Julien I'm not assuming it's even, its just that $f'(0)$ must be equal to $0$ for $f$ not to exceed $1$ in a neighborhood of $0$, and if we can show that $f''(0)=-1$ we would be done. – Olivier Bégassat Mar 4 at 22:50

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@julien : Since $f(0) = 1$, then $f(x) - 1$ has the sign of $f'(0)x$ in a neighborhood of $0$ (but it must remain $\le 0$). – Joel Cohen Mar 4 at 23:25
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## 1 Answer

There is a 1980 paper by John Roe titled A characterization of the sine function, with the following abstract. I suspect the specific result here follows from Roe's theorem.

If a function and all its derivatives and integrals are absolutely uniformly bounded, then the function is a sine function with period 2π.

http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2081636

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Yes, that's probably it. Do you know a link to the full paper? – julien Mar 4 at 21:35
– Steve Kass Mar 4 at 21:42

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If you have a journal subscription, a link to that paper is there. The essence of the approach is to investigate the Fourier transform of $f(x)$. An exponential dampening factor is introduced to make the function $L^1$. – Christopher A. Wong Mar 4 at 21:47
@ChristopherA.Wong Thanks for the sketch of proof. If I asked, it is precisely because I don't have journal subscription... – julien Mar 4 at 21:50
– Git Gud Mar 4 at 21:59
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