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
12 events
| when toggle format | what | by | license | comment | |
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| Jan 19, 2018 at 17:23 | comment | added | Pietro Majer | @David E Speyer, yes, I also noticed that. | |
| Jan 19, 2018 at 16:25 | history | edited | Pietro Majer | CC BY-SA 3.0 |
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| Jan 19, 2018 at 15:47 | comment | added | Pedro Lauridsen Ribeiro | This is a neat counterexample - thanks David! | |
| Jan 19, 2018 at 15:19 | comment | added | David E Speyer | I uploaded some figures in a separate answer. | |
| Jan 19, 2018 at 15:01 | comment | added | David E Speyer | In one variable, if $f(0)=f'(0)=0$ and $f''(0) =c>0$, then I get that $g_n''(0)=c$ for all $n$, so $g''_n$ does not approach $0$. | |
| Jan 19, 2018 at 10:22 | comment | added | Pietro Majer | I made a cleaner version. As to $C^2$ convergence, I'm not sure. I think $\partial_{ij}g_n$ converges uniformly to zero on any $\{f\ge\epsilon\}$ at least. | |
| Jan 19, 2018 at 10:15 | history | edited | Pietro Majer | CC BY-SA 3.0 |
cleaner proof
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| Jan 18, 2018 at 15:01 | history | edited | Pietro Majer | CC BY-SA 3.0 |
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| Jan 18, 2018 at 13:55 | comment | added | Pedro Lauridsen Ribeiro | It could be the case that $C^2$ convergence fails in certain cases. Bony et alii (hehe) showed in the paper cited in my question that if $n\geq 4$ there is a non-negative smooth function in $\mathbb{R}^n$ which is not a finite sum of squares of $C^2$ functions. Perhaps by mimicking their argument in a way similar to David Speyer's answer one could show that $C^2$ convergence of a would-be series of squares fails for the same counterexample. | |
| Jan 18, 2018 at 11:07 | history | edited | Pietro Majer | CC BY-SA 3.0 |
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| Jan 18, 2018 at 7:11 | history | edited | Pietro Majer | CC BY-SA 3.0 |
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| Jan 18, 2018 at 0:32 | history | answered | Pietro Majer | CC BY-SA 3.0 |