Timeline for A moment problem on $[0,1]$ in which infinitely many moments are equal
Current License: CC BY-SA 3.0
12 events
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
Commonmark migration
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| Aug 27, 2013 at 0:11 | vote | accept | Santhosh Kumar | ||
| Aug 26, 2013 at 20:18 | comment | added | Davide Giraudo | @YemonChoi You are perfectly right, $\mu$ and $\nu$ are unneeded. | |
| Aug 26, 2013 at 20:17 | history | edited | Davide Giraudo | CC BY-SA 3.0 |
correct the argument.
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| Aug 26, 2013 at 20:08 | comment | added | Yemon Choi | Moreover, I don't think the chain of logic in your last paragraph is quite right. I don't understand why you are fixing $\mu$ and $\nu$ beforehand. What you want to prove is: if $\sum_{n\in S} n^{-1}<\infty$ then there exist two distinct probability measures $\mu$ and $\nu$ such that $\mu$ and $\nu$ have the same moments on $S$. As Martin and George have pointed out, this follows from taking the two parts of the Hahn-Jordan decomposition of an annihilating measure. | |
| Aug 26, 2013 at 19:35 | comment | added | Davide Giraudo | @YemonChoi Indeed, my initial argument was not quite accurate. Thank you for point this out. I've edited. | |
| Aug 26, 2013 at 19:34 | history | edited | Davide Giraudo | CC BY-SA 3.0 |
added 51 characters in body
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| Aug 26, 2013 at 16:59 | comment | added | Yemon Choi | @Martin thanks. (George Lowther suggested exactly the same, indepdendently, in comments to the main question) | |
| Aug 26, 2013 at 16:54 | comment | added | Martin | @YemonChoi: I didn't understand that part of the answer either (I think Davide wants $+$ twice). But the following should work: Since $F(1) = 0 = m^+(1) - m^-(1)$, the measures $m^{\pm}$ have the same norm, so we can normalize to find two distinct nonzero probability measures measures whose moments-indexed-by-S agree by the choice of $F$. | |
| Aug 26, 2013 at 16:15 | comment | added | Yemon Choi | In your last sentence, where do $\mu$ and $\nu$ come from? If they are given at the start of the question, and assumed to have the same moments-indexed-by-S, then how do you know $\mu-m^+$ and $\nu+m^{-}$ are prob measures? (This is the part that I wasn't sure about, it is trivial from Hahn-Banach and Hahn-Jordan that you can find two positive measures whose moments agree on S) | |
| Aug 26, 2013 at 10:02 | history | edited | Davide Giraudo | CC BY-SA 3.0 |
deleted 1 characters in body
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| Aug 26, 2013 at 9:05 | history | answered | Davide Giraudo | CC BY-SA 3.0 |