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).

Read more about this topic:  Whitney Extension Theorem

Famous quotes containing the word statement:

    “He that writes to himself writes to an eternal public. That statement only is fit to be made public, which you have come at in attempting to satisfy your own curiosity.”
    —Ralph Waldo Emerson (1803–1882)

    “The new statement will comprise the skepticisms, as well as the faiths of society, and out of unbeliefs a creed shall be formed. For, skepticisms are not gratuitous or lawless, but are limitations of the affirmative statement, and the new philosophy must take them in, and make affirmations outside of them, just as much as must include the oldest beliefs.”
    —Ralph Waldo Emerson (1803–1882)

    “The force of truth that a statement imparts, then, its prominence among the hordes of recorded observations that I may optionally apply to my own life, depends, in addition to the sense that it is argumentatively defensible, on the sense that someone like me, and someone I like, whose voice is audible and who is at least notionally in the same room with me, does or can possibly hold it to be compellingly true.”
    —Nicholson Baker (b. 1957)