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Difference between revisions of "Whitney extension theorem"

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==Whitney data==
 
==Whitney data==
Let $K\Subset U$ be a compact subset of $U$. The ''Whitney data'' (or "smooth function in the sense of Whitney") is the collection of continuous functions  
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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  
 
$$
 
$$
\mathbf f=\{f^\a:K\to\R,\ |\a|\le m\},
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\{f^\a:K\to\R,\ |\a|\le m\},
 
$$
 
$$
which satisfies the compatibility condition that were established above for the partial derivatives: for each multiindex $\a$ and natural $r$ the differences
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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,  
 
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,
 
should be small as specified,
 
$$
 
$$
|R^\a_m(a,x)|=o\Big(|x-a|^{r-|\a|}\Big),\qquad x,a\in K,\ |x-a|\to0.
+
|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}}).
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 +
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 $K$.
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 +
== 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]].
 +
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD 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}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> B. Malgrange, "Ideals of differentiable functions" , Oxford Univ. Press (1966) pp. Chapt. I {{MR|2065138}} {{MR|0212575}} {{ZBL|0177.17902}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J.C. Tougeron, "Ideaux de fonction différentiables" , Springer (1972) pp. Chapt. IV {{MR|0440598}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> M. Golubitsky, "Stable mappings and their singularities" , Springer (1973) pp. 108ff {{MR|0341518}} {{ZBL|0294.58004}} </TD></TR></table>
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{|
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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}}
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|-
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|valign="top"|{{Ref|M}}||valign="top"| B. Malgrange, "Ideals of differentiable functions" , Oxford Univ. Press (1966) pp. Chapt. I {{MR|2065138}} {{MR|0212575}} {{ZBL|0177.17902}}  
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|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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Revision as of 14:09, 30 April 2012

$\def\a{\alpha} \def\b{\beta} \def\p{\partial}$

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 $|x-a|$: $$ \Big|f^{(\a)}(x)-\Big(T_a^r f^{(\a)}\Big)(x)\Big|=o\Big(|x-a|^{r-|\a|}\Big). $$ 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, [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 $K$.

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.

References

[W] H. Whitney, "Analytic extensions of differentiable functions defined in closed sets" Trans. Amer. Math. Soc. , 36 (1934) pp. 63–89 MR1501735 Zbl 0008.24902 Zbl 60.0217.01
[M] B. Malgrange, "Ideals of differentiable functions" , Oxford Univ. Press (1966) pp. Chapt. I MR2065138 MR0212575 Zbl 0177.17902
[N] Narasimhan, R. Analysis on real and complex manifolds. North-Holland Mathematical Library, 35. North-Holland Publishing Co., Amsterdam, 1985. MR0832683
How to Cite This Entry:
Whitney extension theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Whitney_extension_theorem&oldid=25762