View source for Whitney extension theorem

{{Use American English|date = March 2019}}
{{Short description|Partial converse of Taylor's theorem}}
In [[mathematics]], in particular in [[mathematical analysis]], the '''Whitney extension theorem''' is a partial converse to [[Taylor's theorem]].  Roughly speaking, the theorem asserts that if ''A'' is a closed subset of a Euclidean space, then it is possible to extend a given function of ''A'' in such a way as to have prescribed derivatives at the points of ''A''. It is a result of [[Hassler Whitney]].

== Statement ==
A precise statement of the theorem requires careful consideration of what it means to prescribe the derivative of a function on a closed set.  One difficulty, for instance, is that closed subsets of Euclidean space in general lack a differentiable structure.  The starting point, then, is an examination of the statement of Taylor's theorem.

Given a real-valued ''C''<sup>''m''</sup> function ''f''('''x''') on '''R'''<sup>''n''</sup>, Taylor's theorem asserts that for each '''a''', '''x''', '''y''' ∈ '''R'''<sup>''n''</sup>, there is a function ''R''<sub>''α''</sub>('''x''','''y''') approaching 0 uniformly as '''x''','''y''' → '''a''' such that
{{NumBlk|::|<math>f({\mathbf x}) = \sum_{|\alpha|\le m} \frac{D^\alpha f({\mathbf y})}{\alpha!}\cdot ({\mathbf x}-{\mathbf y})^{\alpha}+\sum_{|\alpha|=m} R_\alpha({\mathbf x},{\mathbf y})\frac{({\mathbf x}-{\mathbf y})^\alpha}{\alpha!}</math>|{{EquationRef|1}}}}
where the sum is over [[multi-index|multi-indices]]&nbsp;''α''.

Let ''f''<sub>''α''</sub> = ''D''<sup>''α''</sup>''f'' for each multi-index ''α''.  Differentiating (1) with respect to '''x''', and possibly replacing ''R'' as needed, yields
{{NumBlk|::|<math>f_\alpha({\mathbf x})=\sum_{|\beta|\le m-|\alpha|}\frac{f_{\alpha+\beta}({\mathbf y})}{\beta!}({\mathbf x}-{\mathbf y})^{\beta}+R_\alpha({\mathbf x},{\mathbf y})</math>|{{EquationRef|2}}}}
where ''R''<sub>''α''</sub> is ''o''(|'''x'''&nbsp;&minus;&nbsp;'''y'''|<sup>''m''&minus;|''α''|</sup>) uniformly as '''x''','''y''' → '''a'''.

Note that ({{EquationNote|2}}) may be regarded as purely a compatibility condition between the functions ''f''<sub>α</sub> which must be satisfied in order for these functions to be the coefficients of the Taylor series of the function ''f''.  It is this insight which facilitates the following statement:

'''Theorem.'''  Suppose that ''f''<sub>''α''</sub> are a collection of functions on a closed subset ''A'' of '''R'''<sup>''n''</sup> for all multi-indices α with <math>|\alpha|\le m</math> satisfying the compatibility condition ({{EquationNote|2}}) at all points ''x'', ''y'', and ''a'' of ''A''.  Then there exists a function ''F''('''x''') of class ''C''<sup>''m''</sup> such that:
# ''F'' = ''f''<sub>0</sub> on ''A''.
# ''D''<sup>''α''</sup>''F'' = ''f''<sub>''α''</sub> on ''A''.
# ''F'' is real-analytic at every point of '''R'''<sup>''n''</sup>&nbsp;&minus;&nbsp;''A''.

Proofs are given in the original paper of {{harvtxt|Whitney|1934}}, and in {{harvtxt|Malgrange|1967}}, {{harvtxt|Bierstone|1980}} and {{harvtxt|Hörmander|1990}}.

==Extension in a half space==
{{harvtxt|Seeley|1964}} proved a sharpening of the Whitney extension theorem in the special case of a half space. A smooth function on a half space '''R'''<sup>''n'',+</sup> of points where ''x''<sub>''n''</sub> ≥ 0 is a smooth function ''f'' on the interior ''x''<sub>''n''</sub> for which the derivatives ∂<suP>α</sup> ''f'' [[Continuous extension|extend to continuous functions]] on the half space. On the boundary ''x''<sub>''n''</sub> = 0, ''f'' restricts to smooth function. By [[Borel's lemma]], ''f'' can be extended to a  
smooth function on the whole of '''R'''<sup>''n''</sup>. Since Borel's lemma is local in nature, the same argument shows that if <math>\Omega</math> is a (bounded or unbounded) domain in '''R'''<sup>''n''</sup> with smooth boundary, then any smooth function on the closure of <math>\Omega</math> can be extended to a smooth function on '''R'''<sup>''n''</sup>.

Seeley's result for a half line gives a uniform extension map

:<math>\displaystyle{E:C^\infty(\mathbf{R}^+)\rightarrow C^\infty(\mathbf{R}),}</math>

which is linear, continuous (for the topology of uniform convergence of functions and their derivatives on compacta) and takes functions supported in [0,''R''] into functions supported in [−''R'',''R'']

To define <math>E,</math> set<ref>{{harvnb|Bierstone|1980|p=143}}</ref>

:<math>\displaystyle{E(f)(x)=\sum_{m=1}^\infty a_m f(-b_mx)\varphi(-b_mx) \,\,\, (x < 0),}</math>

where φ is a smooth function of compact support on ''R'' equal to 1 near 0 and the sequences (''a''<sub>''m''</sub>), (''b''<sub>''m''</sub>) satisfy:

* <math>b_m > 0</math> tends to <math>\infty</math>;
* <math>\sum a_m b_m^j = (-1)^j</math> for <math>j \geq 0</math> with the sum absolutely convergent.

A solution to this system of equations can be obtained by taking <math>b_n = 2^n</math> and seeking an [[entire function]]

:<math>g(z)=\sum_{m=1}^\infty a_m z^m</math>

such that <math>g\left(2^j\right) = (-1)^j.</math> That such a function can be constructed follows from the [[Weierstrass factorization theorem|Weierstrass theorem]] and [[Mittag-Leffler theorem]].<ref>{{harvnb|Ponnusamy|Silverman|2006|pp=442–443}}</ref>

It can be seen directly by setting<ref>{{harvnb|Chazarain|Piriou|1982}}</ref>

:<math>W(z)=\prod_{j \ge 1} (1-z/2^j),</math>

an entire function with simple zeros at <math>2^j.</math> The derivatives ''W'' '(2<sup>''j''</sup>) are bounded above and below. Similarly the function

:<math>M(z)=\sum_{j \ge 1} {(-1)^j\over W^\prime(2^j) (z-2^j)}</math>

meromorphic with simple poles and prescribed residues at <math>2^j.</math>

By construction

:<math>\displaystyle{g(z)=W(z)M(z)}</math>

is an entire function with the required properties.

The definition for a half space in '''R'''<sup>''n''</sup> by applying the operator ''R'' to the last variable ''x''<sub>''n''</sub>. Similarly, using a smooth [[partition of unity]] and a local change of variables, the result for a half space implies the existence of an analogous extending map

:<math>\displaystyle{C^\infty(\overline{\Omega}) \rightarrow C^\infty(\mathbf{R}^n)}</math>

for any domain <math>\Omega</math> in '''R'''<sup>''n''</sup> with smooth boundary.

== See also ==

* The [[Kirszbraun theorem]] gives extensions of Lipschitz functions.
* {{annotated link|Tietze extension theorem}}
* {{annotated link|Hahn–Banach theorem}}

==Notes==
{{reflist|2}}

==References==
*{{citation | title=Extension of range of functions | first=Edward James | last=McShane | journal=Bull. Amer. Math. Soc. | volume=40 | issue=12 | pages=837–842 | year=1934 | mr=1562984 | zbl=0010.34606 | doi=10.1090/s0002-9904-1934-05978-0| doi-access=free }}
*{{citation|title=Analytic extensions of differentiable functions defined in closed sets|first=Hassler|last=Whitney|authorlink=Hassler Whitney|journal=Transactions of the American Mathematical Society|year=1934|volume=36|pages=63–89|doi=10.2307/1989708|jstor=1989708|issue=1|publisher=American Mathematical Society|doi-access=free}}
*{{citation|journal=Bulletin of the Brazilian Mathematical Society|volume=11|issue=2|year=1980|pages= 139–189| 
title=Differentiable functions|first=Edward|last= Bierstone|authorlink=Edward Bierstone|doi=10.1007/bf02584636}}
*{{citation|last=Malgrange|first= Bernard|title= Ideals of differentiable functions|series= Tata Institute of Fundamental Research Studies in Mathematics|volume=3| publisher=Oxford University Press|year=1967}}
*{{citation|last=Seeley|first= R. T.|title= Extension of C∞ functions defined in a half space|journal=Proc. Amer. Math. Soc. |volume=15|year= 1964 |pages=625–626|doi=10.1090/s0002-9939-1964-0165392-8|doi-access=free}}
*{{citation|last=Hörmander|first= Lars|title= The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis|publisher= Springer-Verlag|year=1990|isbn=3-540-00662-1}}
*{{citation|title=Introduction to the Theory of Linear Partial Differential Equations|volume=14|series= Studies in Mathematics and Its Applications|first=Jacques|last= Chazarain|first2= Alain|last2= Piriou|publisher=Elsevier|year= 1982|isbn=0444864520}}
*{{citation|last=Ponnusamy|first= S.|last2= Silverman|first2= Herb|title= Complex variables with applications|publisher=Birkhäuser|year=2006|isbn= 0-8176-4457-1}}
*{{citation|last=Fefferman|first=Charles|authorlink=Charles Fefferman|title=A sharp form of Whitney's extension theorem|journal=[[Annals of Mathematics]]|year=2005|pages=509–577|doi=10.4007/annals.2005.161.509|mr=2150391|volume=161|issue=1|doi-access=free}}

[[Category:Theorems in analysis]]