View source for Whitney extension theorem

{{MSC|26E10}}$\def\a{\alpha}
\def\b{\beta}
\def\p{\partial}$

A deep theorem from the real analysis, showing which data are required to extent a real-valued function from a compact subset in $\R^n$ to its open neighborhood in a $C^m$-smooth or $C^\infty$-smooth way.

==Jets and terminology==
If $U$ is an open subset in $\R^n$ and $f:U\to\R$ is a smooth function, then one can define its partial derivatives to any order not exceeding the smoothness: in the [[multi-index notation]] the collection of all derivatives
$$
f^{(\a)}=\p^\a f\in C^{m-|\a|}(U),\qquad 0\le |\a|\le m,\ f^{(0)}=f,
$$
is called an $m$-[[jet]] of the function $f\in C^m(U)$.

The different derivatives are related by the obvious formulas $\p^\b f^{(\a)}=f^{(\a+\b)}$ as long as $|\a|+|\b|\le m$. This allows to compare them using the Taylor expansion. For each point $a\in U$ and each derivative $f^{(\a)}$ one can form the Taylor polynomial of order $r\le m-|\a|$ centered at $a$,
$$
\Big(T_a^r f^{(\a)}\Big)(x)=\sum_{|\b|\le r}\frac1{\b!}\Big(\p^\b f^{(\alpha)}(a)\Big)\cdot(x-a)^{\b}=
\sum_{|\b|\le r}\frac1{\b!}f^{(\a+\b)}(a)\cdot(x-a)^{\b}.\tag T
$$
The difference between $f^{(\a)}(x)$ and the value provided by the Taylor polynomial $\Big(T_a^r f^{(\a)}\Big)(x)$ should be small together with $|x-a|$: 
$$
\Big|f^{(\a)}(x)-\Big(T_a^r f^{(\a)}\Big)(x)\Big|=o\Big(|x-a|^{r}\Big).
$$
These asymptotic conditions are necessary for the functions $f^{(\a)}$ to be partial derivatives of a smooth function.

==Whitney data==
Let $K\Subset \R^n$ be a compact subset of $\R^n$. The ''Whitney data'' (or "smooth function in the sense of Whitney") is the collection of continuous functions 
$$
\{f^\a:K\to\R,\ |\a|\le m\},
$$
which satisfies the compatibility condition that were established above for the partial derivatives: for each multiindex $\a$ the differences
$$
R_m^\a(a,x)=f^\a(x)-\sum_{|\b|\le m-|\a|}\frac1{\b!}f^{\a+\b}(a)\cdot(x-a)^\b,\qquad x,a\in K, 
\tag I
$$
should be small as specified,
$$
|R^\a_m(a,x)|=o\Big(|x-a|^{m-|\a|}\Big),\qquad x,a\in K,\ |x-a|\to0.
$$

'''Theorem''' (H. Whitney, 1934, {{Cite|W}}).

Any Whitney data collection on $K\Subset U$ can be extended as a $C^m$-smooth function on $\R^n$. The infinite collection of Whitney data (defined for all $m$) extends as a $C^\infty$-smooth function on $\R^n$. In both cases this means that there exists a smooth function $f:\R^n\to\R$ such that for any multiindex $\a$ the restriction of $f^{(\a)}=\p^\a f$ coincides with the specified $f^\a$ after restriction on $K$.

The proof of this result can be found in {{Cite|M|Ch. 1}}, see also {{Cite|N|Sect. 1.5}}.

== Borel theorem ==
A particular case of the Whitney extension theorem corresponds to $K=\{0\}\Subset\R^n$ being a single point at the origin. In this case the Whitney data reduces to the (finite or infinite) collection of real numbers $c_\a$. The  "integrability conditions" for this special case are void, thus any formal power series $\sum_{\a}c_\a x^\a$ is the Taylor series of some $C^\infty$-smooth function (clearly, the case of finite $m$ is trivial for such compact). This statement is known as the [[Borel theorem]], {{Cite|N|Sect. 1.5}}. 

====References====
{|
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|valign="top"|{{Ref|W}}||valign="top"|H. Whitney, ''Analytic extensions of differentiable functions defined in  closed sets'', Trans. Amer. Math. Soc. , '''36''' (1934) pp. 63–89  {{MR|1501735}} {{ZBL|0008.24902}} {{ZBL|60.0217.01}}
|-
|valign="top"|{{Ref|M}}||valign="top"| B. Malgrange, ''Ideals of differentiable functions'', Oxford Univ. Press  (1966), {{MR|2065138}} {{MR|0212575}} {{ZBL|0177.17902}} 
|-
|valign="top"|{{Ref|N}}||valign="top"|Narasimhan, R.  ''Analysis on real and complex manifolds'', North-Holland Mathematical Library, '''35'''. North-Holland Publishing Co., Amsterdam,  1985. {{MR|0832683}}
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|}