Whitney Extension Theorem - Statement

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 Cm function f(x) on Rn, Taylor's theorem asserts that for each a, x, y ∈ Rn, it is possible to write

(1)

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

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

(2)

where Rα 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 fα 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 fα are a collection of functions on a closed subset A of Rn for all multi-indices α with satisfying the compatibility condition (2) at all points x, y, and a of A. Then there exists a function F(x) of class Cm such that:

  1. F=f0 on A.
  2. DαF = fα on A.
  3. F is real-analytic at every point of Rn-A.

Proofs are given in the original paper of Whitney (1934), as well as in Malgrange (1967), Bierstone (1980) and Hörmander (1990).

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