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Proceedings of the American Mathematical Society
ISSN 1088-6826(e) ISSN 0002-9939(p)

     

On the algebra of functions $\mathcal{C}^k$-extendable for each $k$ finite

Author(s): Wieslaw Pawlucki
Journal: Proc. Amer. Math. Soc. 133 (2005), 481-484.
MSC (2000): Primary 26E10; Secondary 32S05, 32B20
Posted: September 8, 2004
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Abstract | References | Similar articles | Additional information

Abstract: For each positive integer $l$ we construct a $\mathcal C^l$-function of one real variable, the graph $\Gamma$ of which has the following property: there exists a real function on $\Gamma$ which is $\mathcal C^k$-extendable to $\mathbb{R} ^2$, for each $k$ finite, but it is not $\mathcal C^{\infty}$-extendable.

References:

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E. Bierstone, P. D. Milman and W. Paw\lucki, Composite differentiable functions, Duke Math. J. 83 (1996), 607-620. MR 1390657 (97k:32011)

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D. Gokhman, Functions in a Hardy field not ultimately $\mathcal{C}^{\infty }$, Complex Variables Theory Appl. (1) 32 (1997), 1-6. MR 1448476 (98e:26024)

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J. Merrien, Prolongateurs de fonctions différentiables d'une variable réelle, J. Math. Pures Appl. (9) 45 (1966), 291-309. MR 0207937 (34:7750)

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W. Paw\lucki, Examples of functions $\mathcal{C}^{k}$-extendable for each $k$ finite, but not $\mathcal{C}^{\infty }$-extendable. Singularities Symposium - \Lojasiewicz 70, Banach Center Publ. Polish Acad. Sci., Warsaw 44 (1998), 183-187. MR 1677379 (99m:32008)

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H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Am. Math. Soc. 36 (1934), 63-89. MR 1501735

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Additional Information:

Wieslaw Pawlucki
Affiliation: Instytut Matematyki, Uniwersytetu Jagiellonskiego, ul. Reymonta 4, 30-059 Kraków, Poland
Email: Wieslaw.Pawlucki@im.uj.edu.pl

DOI: 10.1090/S0002-9939-04-07756-1
PII: S 0002-9939(04)07756-1
Keywords: $\mathcal C^k$-function, extension, Whitney field
Received by editor(s): October 13, 2003
Posted: September 8, 2004
Additional Notes: This research was partially supported by the KBN grant 5 PO3A 005 21 and the European Community IHP-Network RAAG (HPRN-CT-2001-00271)
Communicated by: David Preiss
Copyright of article: Copyright 2004, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




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