Given an arbitrary set $E \subset \mathbb R^n$, $n\ge 2$, and a function
$f: E \rightarrow \mathbb R$, consider the problem of extending $f$ to a
$C^1$ function defined on the entire $\mathbb R^n$. A procedure for
determining whether such an extension exists was suggested in 1958
by G. Glaeser. In 2004 C. Fefferman proposed a related procedure
for dealing with the much more difficult cases of higher
smoothness. The procedures in question require iterated
computations of some bundles until the bundles stabilize. How many
iterations are needed? We give a sharp estimate for the number of
iterations that could be required in the $C^1$ case. Some related
questions are discussed.
Publié le : 2007-04-14
Classification:
Whitney problems,
extension of smooth functions,
Glaeser refinements,
46E15,
46E35,
46B99
@article{1190831223,
author = {Klartag, Bo'az and Zobin, Nahum},
title = {$C^1$ extensions of functions and stabilization of
Glaeser refinements},
journal = {Rev. Mat. Iberoamericana},
volume = {23},
number = {1},
year = {2007},
pages = { 635-669},
language = {en},
url = {http://dml.mathdoc.fr/item/1190831223}
}
Klartag, Bo'az; Zobin, Nahum. $C^1$ extensions of functions and stabilization of
Glaeser refinements. Rev. Mat. Iberoamericana, Tome 23 (2007) no. 1, pp. 635-669. http://gdmltest.u-ga.fr/item/1190831223/