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 off A in such a way as to have prescribed derivatives at the points of A. It is a result of Hassler Whitney. A related result is due to McShane, hence it is sometimes called the McShane–Whitney extension theorem.

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 Rⁿ, Taylor's theorem asserts that for each a, x, y ∈ Rⁿ, it is possible to write

$f({\bold x}) = \sum_{|\alpha|\le m} \frac{D^\alpha f({\bold y})}{\alpha!}\cdot ({\bold x}-{\bold y})^{\alpha}+\sum_{|\alpha|=m} R_\alpha({\bold x},{\bold y})\frac{({\bold x}-{\bold y})^\alpha}{\alpha!}$ (1)

where α is a multi-index and R_α(x,y) → 0 uniformly as x,y → a.

Let f_α=D^αf for each multi-index α. Differentiating (1) with respect to x, and possibly replacing R as needed, yields

$f_\alpha({\bold x})=\sum_{|\beta|\le m-|\alpha|}\frac{f_{\alpha+\beta}({\bold y})}{\beta!}({\bold x}-{\bold y})^{\beta}+R_\alpha({\bold x},{\bold y})$ (2)

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 Rⁿ for all multi-indices α with $|\alpha|\le m$ 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:

1. F=f₀ on A.
2. D^αF = f_α on A.
3. F is real-analytic at every point of Rⁿ-A.

References

• Extension of range of functions, Edward James McShane, Bull. Amer. Math. Soc., 40:837-842, 1934. MR1562984
• Whitney, Hassler (1934), "Analytic extensions of functions defined in closed sets", Transactions of the American Mathematical Society (American Mathematical Society) 36 (1): 63–89, doi:10.2307/1989708, JSTOR 1989708