If $U$ is a bounded domain in $\mathbb R^n$ whose boundary is smooth, and $f$ is a smooth function on $U$ whose partial derivatives of all orders have a continuous extensions to $\bar U$. For an arbitrary domain $V \supseteq \bar U$, is there a smooth function $\tilde f$ on $V$ extending $f$?
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Why doesn't $e^{-1/x}$ on $(0,\infty)$ serve as a counterexample? – Apr 15 '12 at 07:44
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@William: we can extend $e^{-1/x}$ to $0$ on the left of $0$. – Summer Apr 15 '12 at 09:22
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Smoothly? How? The derivative develops a singularity to the left of zero, no? – Apr 15 '12 at 09:31
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1If $f(x) = {e^{ - 1/x}}$ for $x>0$ and $f=0$ for $x \leq 0$, $f_ + ^{(n)}(0) = 0 = f_ - ^{(n)}(0)$ for all $n=0,1,2,...$, so it is smooth. – Summer Apr 15 '12 at 10:00
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Sure, that works. – Apr 15 '12 at 20:29
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We can take $V=\mathbb R^n$ without losing anything. The answer is yes, but this is nontrivial and I'm not going to prove it here. Here are some sources:
1) Short paper by Seeley (1964) covers the case of half-space. If you are interested in local matters, then straighten out a piece of $\partial U$ and apply this reflection-based extension. http://www.ams.org/journals/proc/1964-015-04/S0002-9939-1964-0165392-8/home.html
2) Whitney's classic paper of 1934 treats general closed sets (!) but is not an easy reading. http://www.ams.org/journals/tran/1934-036-01/S0002-9947-1934-1501735-3/home.html
3) In between there was a paper by M.R. Hestenes, "Extension of the range of a differentiable function" (Duke Math. J. 8, (1941) 183--192). Unfortunately, Duke is less generous with old articles than the AMS; the article is behinds a paywall.
4) A very recent 2-volume book by A. Brudnyi and Yu. Brudnyi "Methods of Geometric Analysis in Extension and Trace Problems" is an encyclopedia of extension theorems. See Chapter 2 of volume 1.
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6There is an IMHO clearer exposition of Whitney's extension theorem in Stein's Singular Integrals and Differentiability Properties of Functions. There's also some renewed interest in this problem due to recent work by Fefferman and Klartag (see this Bulletin paper for an exposition) which has application to fitting of curves/surfaces to data. – Willie Wong Jun 19 '12 at 10:13