Differentiable functions defined in closed sets. A problem of Whitney

In 1934, Whitney raised the question of how to recognize whether a function f defined on a closed subset X of ^sup n^ is the restriction of a function of class ðoe'z^sup p^. A necessary and sufficient criterion was given in the case n=1 by Whitney, using limits of finite differences, and in the...

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Bibliographic Details
Published in:Inventiones mathematicae 2003-02, Vol.151 (2), p.329-352
Main Authors: Bierstone, Edward, Milman, Pierre D., Pawłucki, Wiesław
Format: Article
Language:English
Subjects:
Bundles
Constrictions
Criteria
Functions (mathematics)
Mathematical analysis
Operators
Studies
Tangents
Topology
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Summary:In 1934, Whitney raised the question of how to recognize whether a function f defined on a closed subset X of ^sup n^ is the restriction of a function of class ðoe'z^sup 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^sub X^(p) such that the criterion of order q implies that f is ðoe'z^sup p^. The result gives a new approach to higher-order tangent bundles (or bundles of differential operators) on singular spaces.[PUBLICATION ABSTRACT]
ISSN:0020-9910
1432-1297
DOI:10.1007/s00222-002-0255-6