Timeline for Polynomials with prescribed points to match prescribed bounds
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
Commonmark migration
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| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| S Dec 29, 2013 at 20:57 | history | bounty ended | Dirk | ||
| S Dec 29, 2013 at 20:57 | history | notice removed | Dirk | ||
| Dec 26, 2013 at 17:28 | vote | accept | Dirk | ||
| Dec 24, 2013 at 16:54 | answer | added | user44143 | timeline score: 4 | |
| S Dec 21, 2013 at 22:45 | history | bounty started | Dirk | ||
| S Dec 21, 2013 at 22:45 | history | notice added | Dirk | Draw attention | |
| Nov 7, 2011 at 12:06 | history | edited | Dirk | CC BY-SA 3.0 |
added 2 characters in body
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| Nov 7, 2011 at 8:32 | history | edited | Dirk | CC BY-SA 3.0 |
corrected title.
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| Nov 7, 2011 at 7:24 | history | edited | Dirk | CC BY-SA 3.0 |
Exntended the question, added link
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| Nov 4, 2011 at 8:56 | comment | added | Dirk | Oh, there is a simpler argument: Since there are piecewise monotone interpolating polynomials we can use them directly to interpolate the $x^+$ and $x^-$ (and possibly adding zero interpolation points inbetween). | |
| Nov 4, 2011 at 7:58 | comment | added | Dirk | Such a polynomial always exists: Take a Chebycheff polynomial $P$ of degree high enough (such that is attains the values 1 and -1 in the right order at point $t_i$. Then choose a monotone polynomial $Q$ which maps the points $t_i$ to the prescribed $x_i$ and take $P\circ Q$. Such monotone polynomial interpolants exit by a theorem of Young (ams.org/mathscinet-getitem?mr=0212455). | |
| Nov 3, 2011 at 22:13 | comment | added | user6976 | @Dirk and @fedia: why does such a polynomial exist for every choice of $x^+, x^-$? Say, what is the polynomial if $x_1^+=1/5, x_2^+=1/4, x_1^-=1/3, x_2^-=1/2$? | |
| Nov 3, 2011 at 13:35 | comment | added | Dirk | That is pretty clear. I am willing to add more assumptions and not hoping for a very clean answer. | |
| Nov 3, 2011 at 13:17 | comment | added | fedja | You understand that the answer depends heavily on the location of the points and, thereby, is by no means as simple and clean as in the non-negative case, right? | |
| Nov 3, 2011 at 12:58 | history | asked | Dirk | CC BY-SA 3.0 |