Difference between revisions of "Whitney extension theorem"
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| + | {{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 $f\in C^m(U)$. | ||
| − | and | + | 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}}. | ||
| − | + | == 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 {{Cite|BS}}, {{Cite|F}} and references therein for further information. | |
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| − | |||
| − | |||
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====References==== | ====References==== | ||
| − | + | {| | |
| + | |- | ||
| + | |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}} | ||
| + | |- | ||
| + | |valign="top"|{{Ref|BS}}||valign="top"| 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, {{MR|1814079}} | ||
| + | |- | ||
| + | |valign="top"|{{Ref|F}}||valign="top"| Fefferman, C. ''A sharp form of Whitney's extension theorem'', Ann. of Math. (2) '''161''' (2005), no. 1, 509–577. {{MR|2150391}} | ||
| + | |- | ||
| + | |} | ||
Latest revision as of 12:24, 12 December 2020
2020 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=24595
Whitney extension theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Whitney_extension_theorem&oldid=24595