# 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 (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 $|\alpha |\leq m$ satisfying the compatibility condition (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 Whitney (1934), and in Malgrange (1967), Bierstone (1980) and Hörmander (1990).

## Extension in a half space

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 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
    $\displaystyle {E:C^{\infty }(\mathbf {R} ^{+})\rightarrow C^{\infty }(\mathbf {R} ),}$
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 $E,$ set
    $\displaystyle {E(f)(x)=\sum _{m=1}^{\infty }a_{m}f(-b_{m}x)\varphi (-b_{m}x)\,\,\,(x<0),}$
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
    $g(z)=\sum _{m=1}^{\infty }a_{m}z^{m}$
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
    $W(z)=\prod _{j\geq 1}(1-z/2^{j}),$
an entire function with simple zeros at $2^{j}.$ The derivatives W '(2$^j$) are bounded above and below. Similarly the function
    $M(z)=\sum _{j\geq 1}{(-1)^{j} \over W^{\prime }(2^{j})(z-2^{j})}$
meromorphic with simple poles and prescribed residues at $2^{j}.$
By construction
    $\displaystyle {g(z)=W(z)M(z)}$
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
    $\displaystyle {C^{\infty }({\overline {\Omega }})\rightarrow C^{\infty }(\mathbf {R} ^{n})}$
for any domain $\Omega$ in R$^n$ with smooth boundary.