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Let $c\in\mathbb{R}$ such that $|c|>1$. Does there exist functions $f\in\mathscr{C} ^\infty(\mathbb{R})$ such that $\forall x\in\mathbb{R}, \; f'(x) =f(cx)$ and $f\neq 0$ ? If yes, can we find all those functions ?

I have tried to search analytic such $f$, but the condition $|c|>1$ tells that no such functions exist.

I have tried to work with the Taylor expansion of a solution if there exists one...without success.

Thank you for your help.

P.Fazioli
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  • @MartinR The accepted answer in that link shows that the Taylor series cannot converge. But that doesn't mean there is no $C^{\infty}$ function satisfying the given property, right? – Kavi Rama Murthy Mar 09 '22 at 10:03
  • Related questions: https://math.stackexchange.com/q/445951/42969, https://math.stackexchange.com/q/3626312/42969 – found with Approach0 – Martin R Mar 09 '22 at 10:06
  • @KaviRamaMurthy: Yes, you are right (and I intentionally did not cast a duplicate vote because I had only a quick look at those threads). I have replaced that comment with another one. I would assume that this also has been asked and answered before, but have not found an identical question so far. – Martin R Mar 09 '22 at 10:08
  • @KaviRamaMurthy: What about this answer? https://math.stackexchange.com/a/445965/42969 – Martin R Mar 09 '22 at 10:20
  • @MartinR Only draw back is it defines $f$ only on $(0,\infty)$. – Kavi Rama Murthy Mar 09 '22 at 10:22
  • @MartinR : if moreover $f(0)=0$ is required? – P.Fazioli Mar 09 '22 at 10:23
  • @P.Fazioli: If $f(0) = 0$ and $0 < c < 1$ then $f$ is identically zero: https://math.stackexchange.com/a/3453759/42969. – But that will probably not help here for the case $|c| > 1$. – Martin R Mar 09 '22 at 10:25
  • just a side note: analytic or smooth? In general the notation $C^{\infty }$ is used for smooth functions. For analytic ones is used $C^{\omega }$ instead – Masacroso Mar 09 '22 at 12:03
  • @Masacroso : I mean $f$ smooth, i.e. $f^{(k)}$ exists for every $k$. The problem has no analytic solution $f$ if $|c|>1$. – P.Fazioli Mar 09 '22 at 12:57

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