Differentiable functions defined in closed sets. A problem of Whitney

Abstract

Abstract. In 1934, Whitney raised the question of how to recognize whether a function f defined on a closed subset X of R n is the restriction of a function of class C p . A necessary and sufficient criterion was given in the case n = 1 by Whitney, using limits of finite differences, and in the case p = 1 by Glaeser (1958), using limits of secants. We introduce a necessary geometric criterion, for general n and p, involving limits of finite differences, that we conjecture is sufficient at least if X has a "tame topology". We prove that, if X is a compact subanalytic set, then there exists q = q X (p) such that the criterion of order q implies that f is C p . The result gives a new approach to higher-order tangent bundles (or bundles of differentiable operators) on singular spaces.

Restrictions of smooth functions to a closed subset

“…condition is satisfied. Hence, by [BMP2], proof of 4.20 (see 3.3 above with Y = X), we see that our theorem is an improvement of [BMP2], 4.19: balls are replaced by the sets Ak of N + 1 points.…”

“…The present author [I] Since Whitney's works, it has been widely known that the theory of smooth functions is closely related to the interpolation theory (e.g. [W2], [G2], [K], [MM], [S], [BMP2], [I]). The reason is that differential properties are not punctual but "molecular" (Glaeser) as seen in the bipunctual inequality used to define Whitney function (see [W1]).…”

“…Then X has the full higher order paratangent spaces at the accumulating point. Remember that if X has full higher order paratangent spaces at any point of X, the conjecture (*) is valid for such X (see [BMP2] 3) A sufficient condition for 2) for general n, d is given in [AS], 3.6. A convenient version of this condition is given in [I] [BMP2]).…”

Fitting a Cm-smooth function to data, I

“…. , * , for the same number * as in Section 2, which is a constant that depends solely on m and n. The algorithm returns the number (2) max{onset A x, * : x ∈ E} which may be easily computed, as was explained in Section 3, and that has the same order of magnitude as max{onset Γ(x, * ) : x ∈ E}. This completes the description of our algorithm.…”

“…The question goes back to Whitney [35], [36], [37], with contri-butions by Glaeser [22], Brudnyi-Shvartsman [5]- [10] and [29], [30], [31], Zobin [38], [39], Bierstone-Milman-Paw lucki [2], [3], Fefferman [13]- [19] and A. and Y. Brudnyi [4]. Here, we take E finite, and we pose the question from the viewpoint of theoretical computer science.…”