Timeline for The closures in $C^0(\mathbb{R}, \mathbb{R})$ of the set of integer valued polynomials, resp, of polynomials with integer coefficients
Current License: CC BY-SA 2.5
16 events
| when toggle format | what | by | license | comment | |
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| Jul 9, 2014 at 17:36 | comment | added | Pietro Majer | (I'll rewrite it more clearly indeed) | |
| Jul 1, 2014 at 17:17 | vote | accept | Pietro Majer | ||
| Jul 1, 2014 at 7:27 | comment | added | Pietro Majer | Hi, what is not clear, exactly? | |
| Jul 1, 2014 at 2:44 | comment | added | joaopa | Hi Pietro, could you give more details about your proof que the closure of $\mathrm{Int}(\mathbb N,\mathbb Z)$ is the set ofcontinuous functions on $\mathbb R$ that take integer values on $\mathbb N$. Thanks in advance. | |
| Apr 6, 2014 at 18:13 | vote | accept | Pietro Majer | ||
| Jul 1, 2014 at 17:17 | |||||
| Dec 31, 2010 at 12:26 | comment | added | paul Monsky | @Laurent: Hi Laurent! I remembered the talk you gave on this at Brandeis years ago. | |
| Dec 31, 2010 at 12:11 | comment | added | Pietro Majer | Très intéressant. Merci Laurent! | |
| Dec 31, 2010 at 10:25 | comment | added | Laurent Berger | @Paul: Hi Paul! I wrote a survey of Ferguson's results which you can find at the very bottom of this page umpa.ens-lyon.fr/~lberger/publications.html (it's in French, sorry). The main results are that if $K$ is a compact subset of $R$ then $Z[X]$ is discrete in $C^0(K,R)$ if the capacity of $K$ is $\geq 1$ and otherwise there is a finite set $J(K) \subset K$ such that a function $f$ is a limit of elements of $Z[X]$ if and only if there exists some $P(X) \in Z[X]$ such that $f(a)=P(a)$ for all $a \in J(K)$. For example if $K=[0,1]$ then $J(K)=\{0,1\}$. | |
| Dec 31, 2010 at 2:14 | comment | added | paul Monsky | Pietro, SJR--I believe the answer is yes and that a proof is to be found in Le Baron O. Ferguson's book: "Approximation by polynomials with integral coefficients", Math Surveys and monographs vol. 17. He also had a survey article in the Monthly on the topic; perhaps there's a proof there as well. | |
| Dec 30, 2010 at 22:23 | comment | added | Pietro Majer | I don't know! very nice question. | |
| Dec 30, 2010 at 11:03 | comment | added | Sidney Raffer | Or maybe Z[x] is closed? | |
| Dec 30, 2010 at 9:16 | comment | added | Sidney Raffer |
No, this is very nice - Thanks for being two. What about the closure of $\mathbb{Z}[x]$?
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| Dec 30, 2010 at 8:59 | history | edited | Pietro Majer | CC BY-SA 2.5 |
added 2 characters in body
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| Dec 30, 2010 at 8:40 | comment | added | Pietro Majer | I see, but let me say that you should think more before posting a question, and avoid these schyzoid dialogues with yourself. | |
| Dec 30, 2010 at 8:36 | comment | added | Pietro Majer | Sorry if the style is somehow redundant; just to be clear. | |
| Dec 30, 2010 at 8:34 | history | answered | Pietro Majer | CC BY-SA 2.5 |