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The Whitney theorem is qualitative, but its quantitative reformulation is of interest for applications. More specifically, given all (or just a part) of Whitney data, one can look for the smooth extension with effective estimate of the $C^m$-norm. In such quantitative setting the problem is interesting and highly nontrivial even for the problem of extension of functions from finite point set.
 
The Whitney theorem is qualitative, but its quantitative reformulation is of interest for applications. More specifically, given all (or just a part) of Whitney data, one can look for the smooth extension with effective estimate of the $C^m$-norm. In such quantitative setting the problem is interesting and highly nontrivial even for the problem of extension of functions from finite point set.
  
The phenomenon can be roughly described as follows: given a finite subset $K\Subset\R^n$ and a function $f^0:K\to\R$ (finite collection of values), one looks for a $C^m$-smooth function $f:\R^n\to\R$ with the explicitly controlled $C^m$-norm, which would interpolate $f^0$, i.e., $f|_K=f^0$. It turns out that there always exists a finite number $N=N(n,m)$, depending only on $n,m$, such that the the norm of the extension $\|f\|_{C^m}$ is sufficient to verify only for extensions from all $N$-point subsets of $K$. For instance, $N(2,2)=6$, and all obstructions to "economic" $C^m$-smooth extension of $f^0$ appear already for 6-point subsets. See {{Cite|BS}}, {{Cite|F}} and references therein for further information.
+
The phenomenon can be roughly described as follows: given a finite subset $K\Subset\R^n$ and a function $f^0:K\to\R$ (finite collection of values), one looks for a $C^m$-smooth function $f:\R^n\to\R$ with the explicitly controlled $C^m$-norm, which would interpolate $f^0$, i.e., $f|_K=f^0$. It turns out that there always exists a finite number $N=N(n,m)$, depending only on $n,m$, such that the norm of the extension $\|f\|_{C^m}$ is sufficient to verify only for extensions from all $N$-point subsets of $K$. For instance, $N(2,2)=6$, and all obstructions to "economic" $C^m$-smooth extension of $f^0$ appear already for 6-point subsets. See {{Cite|BS}}, {{Cite|F}} and references therein for further information.
  
 
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Latest revision as of 12:24, 12 December 2020

2010 Mathematics Subject Classification: Primary: 26E10 [MSN][ZBL]$\def\a{\alpha} \def\b{\beta} \def\p{\partial}$

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, [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 [M, Ch. 1], see also [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, [N, Sect. 1.5].

Quantitative versions

The Whitney theorem is qualitative, but its quantitative reformulation is of interest for applications. More specifically, given all (or just a part) of Whitney data, one can look for the smooth extension with effective estimate of the $C^m$-norm. In such quantitative setting the problem is interesting and highly nontrivial even for the problem of extension of functions from finite point set.

The phenomenon can be roughly described as follows: given a finite subset $K\Subset\R^n$ and a function $f^0:K\to\R$ (finite collection of values), one looks for a $C^m$-smooth function $f:\R^n\to\R$ with the explicitly controlled $C^m$-norm, which would interpolate $f^0$, i.e., $f|_K=f^0$. It turns out that there always exists a finite number $N=N(n,m)$, depending only on $n,m$, such that the norm of the extension $\|f\|_{C^m}$ is sufficient to verify only for extensions from all $N$-point subsets of $K$. For instance, $N(2,2)=6$, and all obstructions to "economic" $C^m$-smooth extension of $f^0$ appear already for 6-point subsets. See [BS], [F] and references therein for further information.

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), 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
[BS] Brudnyi, Y. and Shvartsman, P. Whitney's extension problem for multivariate $C^{1,\omega}$-functions. Trans. Amer. Math. Soc. 353 (2001), no. 6, 2487–2512, MR1814079
[F] Fefferman, C. A sharp form of Whitney's extension theorem, Ann. of Math. (2) 161 (2005), no. 1, 509–577. MR2150391
How to Cite This Entry:
Whitney extension theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Whitney_extension_theorem&oldid=50958