Whitney extension 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

where the sum is over 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

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 $, \alpha, \le m$ 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, ''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 $\Omega$ is a (bounded or unbounded) domain in R^''n'' with smooth boundary, then any smooth function on the closure of $\Omega$ can be extended to a smooth function on R^''n''.

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

:$\displaystyle$

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 $E,$ set

:$\displaystyle$

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

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

A solution to this system of equations can be obtained by taking $b_n = 2^n$ and seeking an entire function

:$g(z)=\sum_^\infty a_m z^m$

such that $g\left(2^j\right) = (-1)^j.$ That such a function can be constructed follows from the Weierstrass theorem and Mittag-Leffler theorem.

It can be seen directly by setting

:$W(z)=\prod_ (1-z/2^j),$

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

:$M(z)=\sum_$

meromorphic with simple poles and prescribed residues at $2^j.$

By construction

:$\displaystyle$

is an entire function with the required properties.

The definition for a half space in R^''n'' by applying the operator ''E'' 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

:$\displaystyle$

for any domain $\Omega$ in R^''n'' with smooth boundary.

See also

* The Kirszbraun theorem gives extensions of Lipschitz functions.
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Notes

References

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*{{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

Theorems in mathematical analysis