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Let $f : M \rightarrow N$ be a holomorphic map between complex manifolds (I'd be interested even in the case $M=N=\mathbb{C}$ which should not be much different).

Now take $K$ a compact subset of $M$, say with no isolated point for the question to be non trivial, and consider the restriction of $f$ to $M$.

How can one recognize intrinsically (ie only looking at the values of $f$ on $K$) that $f_{|K}$ is secretely the restriction of a holomorphic map (in a neighborhood of $K$) ?

Obviously such a restriction has to be locally lipschitz ; but it should also have some stronger, more constraining properties.

Albert
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  • I'm not sure you understand the question I asked (maybe I wasn't very clear). what I want to know is, given a function on $K$ (let's say locally lipschitz), how can I know if it extends to a holomorphic function on a nghbd of $K$ ? obviously very few will allow this – Albert Jul 02 '14 at 21:25
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    Mergelyan's Theorem (here is the Wikipedia article ) suggests that this could be difficult -- any continuous map on $K$ can be uniformly approximated by rational functions. – Charles Baker Jul 02 '14 at 22:17
  • "should not be much different": the case of manifolds of complex dimension $>1$ is very much different indeed. Did you mean 1-dimensional manifolds? –  Jul 02 '14 at 22:21
  • Do you have any particular type of $K$ in mind, something like a sequence with a limit point or something substantial enough to squeeze out directional derivatives? I think form of the answer will very much depend on the 'size' of $K$. – Conifold Jul 02 '14 at 22:30
  • @Conifold a priori not, but a cantor set should be interesting. well even a smooth curve seems non trivial. – Albert Jul 02 '14 at 22:42
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    @user52733 I am aware that the question is not trivial. but since it seems like a natural question, I wondered if there are some known results about this (maybe negative). – Albert Jul 02 '14 at 22:43
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    See this paper... the spaces of holomorphic germs on compact sets were studied for many decades, but it seems that the "classical" results for scalar valued functions just can't be found anymore. These days it absolutely has to be something about locally convex spaces... the curse of generalization. –  Jul 02 '14 at 22:53
  • If the smooth curve is analytic then my guess is the composition with its analytic parametrization has to be real analytic (Taylor series locally converge), those are exactly the functions that extend to holomorphic ones in a neighborhood of the real line. For some fractals there might be enough calculus on them to get some distributional Cauchy-Riemann http://www.ams.org/notices/199910/fea-strichartz.pdf. – Conifold Jul 02 '14 at 22:57
  • @Conifold I suspect you are right concerning analytic curves. I'll have a look at your link, but I am really interested in arbitrary compacts, for which I doubt a version of Weyl's lemma could work. – Albert Jul 02 '14 at 23:04
  • @Thisismuchhealthier. thanks a lot ! it will take me some time to digest, but this seems to answer perfectly my questions ! – Albert Jul 02 '14 at 23:05
  • I guess the keyword I was missing was "jet" – Albert Jul 02 '14 at 23:07

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