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.
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 â R<sup>n</sup>, there is a function R<sub>ñ</sub>(x,y) approaching 0 uniformly as x,y â a such that
where the sum is over multi-indices ñ.
Let f<sub>ñ</sub> = D<sup>ñ</sup>f for each multi-index ñ. Differentiating (1) with respect to x, and possibly replacing R as needed, yields
where R<sub>ñ</sub> is o(|x − y|<sup>m−|ñ|</sup>) uniformly as x,y â a.
Note that () may be regarded as purely a compatibility condition between the functions f<sub>ñ</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>ñ</sub> are a collection of functions on a closed subset A of R<sup>n</sup> for all multi-indices ñ with satisfying the compatibility condition () at all points x, y, and a of A. Then there exists a function F(x) of class C<sup>m</sup> such that:
Proofs are given in the original paper of , and in , and .
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> âÂÂ¥ 0 is a smooth function f on the interior x<sub>n</sub> for which the derivatives âÂÂ<suP>ñ</sup> f 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 is a (bounded or unbounded) domain in R<sup>n</sup> with smooth boundary, then any smooth function on the closure of can be extended to a smooth function on R<sup>n</sup>.
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 [0,R] into functions supported in [âÂÂR,R]
To define set
where ÃÂ 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:
A solution to this system of equations can be obtained by taking and seeking an entire function
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<sup>j</sup>) 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<sup>n</sup> by applying the operator E 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
for any domain in R<sup>n</sup> with smooth boundary.