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Fix any $f\in\mathcal{S}_x(\mathbb{R}^d\to\mathbb{C})$, i.e. $f$ is a Schwartz function from $\mathbb{R}^d$ to $\mathbb{C}$. Is it always possible to find $g,h\in\mathcal{S}_x(\mathbb{R}^d\to\mathbb{C})$ such that $f(x)=g(x)h(x)$ for all $x\in\mathbb{R}^d$?

The square root seems a good choice, but I find this problem. So I am not sure what I am supposed to do to solve or disprove this problem.

Thank you!

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  • $\begingroup$ I think that this should be accomplishable. Where $f$ is bounded away from zero the square root works. For zeros I would suggest a combination of the function itself and the constant $1$ function. Then you need to glue square root and original function respectively constant $1$ and original function together via a suitable partition of unity or so. However you have to be careful since when the function has to much zeros (like one with compact support) you cannot do it in that way but take the product of the function with itself. $\endgroup$ – Sebastian Bechtel Mar 9 '17 at 10:33
  • $\begingroup$ Another suggestion would be Fourier transform. Then you would need something like a convolution unit in the Schwartz space, but not sure whether something like this exists. Sadly delta is not a regular tempered distribution :P $\endgroup$ – Sebastian Bechtel Mar 9 '17 at 10:34
  • $\begingroup$ @SebastianBechtel, one could argue that such a convolution unit would equal the Dirac delta measure (as a tempered distribution), and therefore could not be an $L^{1}$ function. In particular, it could not be in the Schwarz space. $\endgroup$ – fourierwho Jul 9 '18 at 23:22
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The answer is yes.

It is Lemma 1 from the article by K. Miyazaki, Distinguished elements in a space of distributions. J. Sci. Hiroshima Univ. Ser. A 24 (1960), 527–533.

This also done in H. Petzeltová and P. Vrbová, Factorization in the algebra of rapidly decreasing functions on $R_n$. Comment. Math. Univ. Carolin. 19 (1978), no. 3, 489–499 as well as in J. Voigt, Factorization in some Fréchet algebras of differentiable functions. Studia Math. 77 (1984), no. 4, 333–348.

Finally, there is a pedagogical account in the note "Weil-Schwartz envelopes for rapidly decreasing functions" by Paul Garrett.

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  • $\begingroup$ Please try to describe as much here as possible in order to make the answer self-contained. Links are fine as support, but they can go stale and then an answer which is nothing more than a link loses its value. Please read this post. $\endgroup$ – robjohn Jul 9 '18 at 23:30
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    $\begingroup$ If someone is really interested in the answer to this question but is too lazy to look at the note by Garrett which contains a proof of the factorization property, then it's too bad for that someone. I don't have time to say more than what I wrote. $\endgroup$ – Abdelmalek Abdesselam Jul 9 '18 at 23:38
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    $\begingroup$ A problem could be that in five years your link does not work anymore, and "this note" by Paul Garrett is not all that telling. I mean you could at least say the note "Weil-Schwartz envelopes for rapidly decreasing functions." Since you appear to be extremely busy; I'll lend a hand. :-) Thanks for the answer though. I think the collection of precises references is really useful. $\endgroup$ – quid Jul 10 '18 at 23:34

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