Timeline for Extending continuous functions from $\partial X$ to $X\cup \partial X$
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Mar 18, 2016 at 18:31 | comment | added | EM90 | I checked ThomasRot's advice, as NWMT says, Tietze Theorem seems to work! I actually need only the continuity of the extended function (which, actually, takes values in $\mathbb{C}$, but this seems not problematic). Thanks to everyone for the help! | |
| Mar 17, 2016 at 14:19 | comment | added | NWMT | ThomasRot is right, $X \cup \partial X$ is a compact Hausdorff space so the Tietze extension theorem applies to your situation, if you only require continuity. What BenoîtKloeckner is refering to also works but it is much more powerful. | |
| Mar 17, 2016 at 13:29 | comment | added | Benoît Kloeckner | Yes, I think that is what I had in mind (I have not worked with this myself, I just remembered that Besson-Courtois-Gallot have used these measure to prove their celebrated rigidity result). The reference Pacific J. Math. Volume 159, Number 2 (1993), 241-270 by Coornaert seems relevant by I have trouble retrieving it to check. | |
| Mar 17, 2016 at 9:54 | comment | added | EM90 | @BenoîtKloeckner: Are you referring to something similar to the Patterson-Sullivan construction? (I'm not really into it, so maybe I'd better deep my knowledge about it. Thanks for reading, thanks for the advice.) | |
| Mar 17, 2016 at 9:54 | comment | added | EM90 | @ThomasRot: Thanks for the advice, I didn't think about it. I'll work on it. | |
| Mar 16, 2016 at 12:57 | comment | added | Benoît Kloeckner | A possible strategy would be to associate to each point $x\in X$ a probability measure $\mu_x$ on $\partial X$, such that the measure depends continuously on the point and converges to $\delta_\zeta$ when $x$ converges to $\zeta\in\partial X$. I think such construction exist at least in particular cases (e.g. limit of the uniform measure on spheres, or using the critical exponent maybe). Then you extend $f$ by $\int f d\mu_x$. | |
| Mar 16, 2016 at 12:38 | comment | added | Thomas Rot | I don't know anything about the setting, but does the Tietze extension theorem apply here? en.wikipedia.org/wiki/Tietze_extension_theorem | |
| Mar 16, 2016 at 11:15 | history | asked | EM90 | CC BY-SA 3.0 |