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I'm looking into the specific case of sets of Real functions closed under derivatives that don't contain zero such as {sin,cos,-sin,-cos}. Is there any information on such sets?

jgon
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Uri George Peterzil
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  • Never heard anything specific. Are you only interested in finite sets? – Arnaud Mortier Mar 21 '18 at 22:05
  • @DietrichBurde No, that one includes $0$. The singleton ${\exp}$ doesn't, though. – Arnaud Mortier Mar 21 '18 at 22:05
  • I don't quite understand your question, surely you could just take any set of functions and remove zero if you need to? – AdJoint-rep Mar 21 '18 at 22:06
  • @AdJoint-rep It wouldn't be closed then (e.g. if you contain one polynomial and you are closed then you have to contain $0$). – Arnaud Mortier Mar 21 '18 at 22:07
  • https://math.stackexchange.com/questions/963180/the-finite-dimensional-function-spaces-that-are-closed-under-taking-derivative https://math.stackexchange.com/questions/945041/finite-dimensional-spaces (contain zero by definition since they are vector spaces, but reading the answers should help to characterize those) – Clement C. Mar 21 '18 at 22:09
  • You're right, I didn't read the description of the sets you need in the title of the question. Sorry. – AdJoint-rep Mar 21 '18 at 22:10
  • It seems that you're looking at differential ideals of the ring $(C^\infty(\Bbb R), D)$, where $D\colon C^\infty(\Bbb R) \to C^\infty(\Bbb R)$ is given by $D(f)\doteq f'$, but without the zero element of the ring. I'm not sure if there is anything interesting about such structures, though. – Ivo Terek Mar 21 '18 at 22:11
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    The only functions that after taking a finite number of derivatives becomes $0$ are the polynomials. Therefore $C^{\infty}(\mathbb{R})\setminus \mathbb{C}[x]$ is the maximal such set. – SphericalTriangle Mar 21 '18 at 22:15

1 Answers1

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Take any continuously differentiable real function that is not a polynomial and take the set of all derivatives of this function.

This set will not contain zero since only constant functions have zero derivatives, and only polynomials eventually differentiate to constants. By definition it is closed under differentiation.

All the sets that you are describing will be unions of sets constructed in this way. I don't think these sets have a name.

AdJoint-rep
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