In
mathematics, in particular in
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
These theories are usually studied ...
, the Whitney extension theorem is a partial converse to
Taylor's theorem
In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the t ...
. 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
Hassler Whitney (March 23, 1907 – May 10, 1989) was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, characteristic classes, and geometric integration ...
.
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
where the sum is over
multi-indices
Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of 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
where ''R''
''α'' is ''o''(, x − y,
''m''−, ''α'', ) uniformly as x,y → a.
Note that () 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
satisfying the compatibility condition () 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 , and in , and .
Extension in a half space
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''
extend to continuous functions on the half space. On the boundary ''x''
''n'' = 0, ''f'' restricts to smooth function. By
Borel's lemma
In mathematics, Borel's lemma, named after Émile Borel, is an important result used in the theory of asymptotic expansions and partial differential equations.
Statement
Suppose ''U'' is an open set in the Euclidean space R''n'', and suppose that ...
, ''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
is a (bounded or unbounded) domain in R
''n'' with smooth boundary, then any smooth function on the closure of
can be extended to a smooth function on R
''n''.
Seeley's result for a half line gives a uniform extension map
:
which is linear, continuous (for the topology of uniform convergence of functions and their derivatives on compacta) and takes functions supported in
,''R''into functions supported in
��''R'',''R''
To define
set
:
where φ is a smooth function of compact support on ''R'' equal to 1 near 0 and the sequences (''a''
''m''), (''b''
''m'') satisfy:
*
tends to
;
*
for
with the sum absolutely convergent.
A solution to this system of equations can be obtained by taking
and seeking an
entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any fini ...
:
such that
That such a function can be constructed follows from the
Weierstrass theorem and
Mittag-Leffler theorem.
It can be seen directly by setting
:
an entire function with simple zeros at
The derivatives ''W'' '(2
''j'') are bounded above and below. Similarly the function
:
meromorphic with simple poles and prescribed residues at
By construction
:
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
In mathematics, a partition of unity of a topological space is a set of continuous functions from to the unit interval ,1such that for every point x\in X:
* there is a neighbourhood of where all but a finite number of the functions of are ...
and a local change of variables, the result for a half space implies the existence of an analogous extending map
:
for any domain
in R
''n'' with smooth boundary.
See also
* The
Kirszbraun theorem gives extensions of Lipschitz functions.
*
*
Notes
References
*
*
*
*
*
*
*
*
*{{citation, last=Fefferman, first=Charles, authorlink=Charles Fefferman, title=A sharp form of Whitney's extension theorem, journal=
Annals of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study.
History
The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as th ...
, year=2005, pages=509–577, doi=10.4007/annals.2005.161.509, mr=2150391, volume=161, issue=1, doi-access=free
Theorems in analysis