Timeline for Is every non-negative test function the limit of a sequence of sums of squares of test functions?
Current License: CC BY-SA 3.0
22 events
| when toggle format | what | by | license | comment | |
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| Jan 19, 2018 at 15:49 | comment | added | Pietro Majer | ok I hadn't noticed | |
| Jan 19, 2018 at 15:39 | comment | added | Pedro Lauridsen Ribeiro | @PietroMajer Fefferman and Phong get a finite sum of squares, so the convergence is (trivially) $C^{1,1}$ but not $C^2$ since the squares cannot be $C^2$ by the results of Bony et alii. | |
| Jan 19, 2018 at 15:18 | answer | added | David E Speyer | timeline score: 3 | |
| Jan 19, 2018 at 14:44 | comment | added | Pietro Majer | @PedroLauridsenRibeiro About the quoted result by Fefferman and Phong: precisely, in what topology converges the series of squares? | |
| Jan 18, 2018 at 3:29 | comment | added | Pedro Lauridsen Ribeiro | Thanks @paulgarrett, I wasn't aware of this interesting theorem. Unfortunately, a closer look (as e.g. in math.ubc.ca/~cass/research/pdf/Dixmier-Malliavin.pdf) doesn't reveal how helpful it is for my question - it doesn't seem at all guaranteed that the rhs of the Dixmier-Malliavin formula applied to this question's context should be a sum of squares. The counterexamples discussed by Bony et alii and in David Speyer's answer below rather indicate that the Dixmier-Malliavin theorem is most likely not relevant to the problem at hand. | |
| Jan 18, 2018 at 1:08 | comment | added | paul garrett | One might also think of the Dixmier-Malliavin theorem that answers analogous questions on real Lie groups, etc. | |
| Jan 18, 2018 at 0:32 | answer | added | Pietro Majer | timeline score: 5 | |
| Dec 28, 2017 at 10:48 | answer | added | David E Speyer | timeline score: 7 | |
| Dec 28, 2017 at 3:09 | history | edited | Pedro Lauridsen Ribeiro | CC BY-SA 3.0 |
Typo corrected
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| Dec 24, 2017 at 0:21 | comment | added | Pedro Lauridsen Ribeiro | I've just added the follow-up subquestion you suggested. I don't know, though, if it's good form for MO to do it since I've already accepted Pietro Majer's answer, or if it is better asked as a separate question because of that... | |
| Dec 24, 2017 at 0:19 | history | edited | Pedro Lauridsen Ribeiro | CC BY-SA 3.0 |
Added follow-up subquestion suggested in comments
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| Dec 24, 2017 at 0:14 | comment | added | André Henriques | Yes, it's going to be much harder... but only because Pietro Majer's answer was so easy. ;-) | |
| Dec 23, 2017 at 23:43 | comment | added | Pedro Lauridsen Ribeiro | I think this, if true, is bound to be (much?) harder, specially if we want convergence to take place in $\mathscr{D}(\mathbb{R}^n)$, since the "cheapest" way to get the result I asked for is to keep the approximating sequence away from zero at the zeros of $f$ which are cluster points of $\{f>0\}$, as Pietro Majer proposed below. In your case, one does the opposite. A possibility would be to somehow "flatten" the approximating functions near these troublesome points, but this probably rules out convergence in $\mathscr{D}(\mathbb{R}^n)$ since, again, the derivatives of $f$ may be large there. | |
| Dec 23, 2017 at 22:05 | comment | added | André Henriques | Maybe it would have been more interesting to ask whether every $0\leq f\in\mathscr{D}(\mathbb{R}^n)$ is the increasing limit of a sequence of sums of squares in $\mathscr{D}(\mathbb{R}^n)$? Or, even better, whether it can be written as $f=\sum_{j=1}^\infty g_j^2$. | |
| Dec 23, 2017 at 21:25 | vote | accept | Pedro Lauridsen Ribeiro | ||
| Dec 23, 2017 at 21:11 | answer | added | Pietro Majer | timeline score: 7 | |
| Dec 23, 2017 at 20:56 | history | edited | Pedro Lauridsen Ribeiro | CC BY-SA 3.0 |
Typos fixed
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| Dec 23, 2017 at 3:39 | comment | added | Pedro Lauridsen Ribeiro | Hmm... The difficulty with your idea is that there may be points $x$ where $f(x)=0$ but the derivatives of $f$ at such $x$ may be large (such points are in the interior of $\mathrm{supp} f$, of course). It's not clear to me how to modify $f$ near $f^{-1}(0)$ so as to put it in square form (to do it away from there is easy if we use squares of smooth functions as elements of a suitable partition of unity since there simply taking the square root of $f$ is harmless) while keeping at a finite but arbitrarily large number of derivatives not too far off the derivatives of $f$ near $f^{-1}(0)$. | |
| Dec 23, 2017 at 1:47 | history | edited | Pedro Lauridsen Ribeiro | CC BY-SA 3.0 |
typo removed
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| Dec 23, 2017 at 1:18 | comment | added | Christian Remling | In dimension $n=1$, we can simply modify $f$ slightly near the edges of an interval where $f>0$ to make it a square there, and then we do this for finitely many of these intervals and set $g=0$ otherwise to obtain an approximation. In general, can't we do essentially the same thing with the components of the open set $\{f>0\}$ ? | |
| Dec 23, 2017 at 0:36 | history | edited | Pedro Lauridsen Ribeiro | CC BY-SA 3.0 |
Added remark
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| Dec 22, 2017 at 23:52 | history | asked | Pedro Lauridsen Ribeiro | CC BY-SA 3.0 |