Differentiable functions defined in closed sets. A problem of Whitney

By Edward Bierstone, Pierre D. Milman, W. Pawłucki. 2001

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Abstract.In 1934, Whitney raised the question of how to recognize whether a function f defined on a closed subset X of ℝn is the restriction of a function of class 𝒞p. 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=qX(p) such that the criterion of order q implies that f is 𝒞p. The result gives a new approach to higher-order tangent bundles (or bundles of differential operators) on singular spaces.

Subfields: Applied Mathematics, Computational Mechanics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c9ff026bfcae1495ac3981da93b3c6a6c9a65563