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{{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''''m'' function ''f''('''x''') on '''R'''''n'', Taylor's theorem asserts that for each '''a''', '''x''', '''y''' ∈ '''R'''''n'', there is a function ''R''''α''('''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]] ''α''.

Let ''f''''α'' = ''D''''α''''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''''α'' is ''o''(|'''x''' − '''y'''|''m''−|''α''|) uniformly as '''x''','''y''' → '''a'''.

Note that ({{EquationNote|2}}) may be regarded as purely a compatibility condition between the functions ''f''α 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''''α'' are a collection of functions on a closed subset ''A'' of '''R'''''n'' 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''''m'' such that:
# ''F'' = ''f''0 on ''A''.
# ''D''''α''''F'' = ''f''''α'' on ''A''.
# ''F'' is real-analytic at every point of '''R'''''n'' − ''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'''''n'',+ of points where ''x''''n'' ≥ 0 is a smooth function ''f'' on the interior ''x''''n'' for which the derivatives ∂α ''f'' [[Continuous extension|extend to continuous functions]] on the half space. On the boundary ''x''''n'' = 0, ''f'' restricts to smooth function. By [[Borel's lemma]], ''f'' can be extended to a
smooth function on the whole of '''R'''''n''. 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'''''n'' with smooth boundary, then any smooth function on the closure of <math>\Omega</math> can be extended to a smooth function on '''R'''''n''.

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''''m''), (''b''''m'') 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''j'') 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'''''n'' by applying the operator ''R'' to the last variable ''x''''n''. 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'''''n'' 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]]

Whitney extension theorem - Revision history

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