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.
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.
Given a real-valued Cm function f(x) on Rn, Taylor's theorem asserts that for each a, x, y ∈ Rn, there is a function Rα(x,y) approaching 0 uniformly as x,y → a such that where the sum is over multi-indices α.
Let fα = Dαf for each multi-index α. Differentiating (1) with respect to x, and possibly replacing R as needed, yields where Rα is o(|x − y|m−|α|) uniformly as x,y → a.
Suppose that fα are a collection of functions on a closed subset A of Rn for all multi-indices α with
Then there exists a function F(x) of class Cm such that: Proofs are given in the original paper of Whitney (1934), and in Malgrange (1967), Bierstone (1980) and Hörmander (1990).
Seeley (1964) proved a sharpening of the Whitney extension theorem in the special case of a half space.
set[1] where φ is a smooth function of compact support on R equal to 1 near 0 and the sequences (am), (bm) satisfy: A solution to this system of equations can be obtained by taking
Similarly the function meromorphic with simple poles and prescribed residues at
The definition for a half space in Rn by applying the operator R to the last variable xn.
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