Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Smooth approximation of definable continuous functions

Author: Andreas Fischer
Journal: Proc. Amer. Math. Soc. 136 (2008), 2583-2587
MSC (2000): Primary 03C64; Secondary 26E10
Published electronically: February 29, 2008
MathSciNet review: 2390530
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \mathcal{M}$ be an $ o$-minimal expansion of the real exponential field which possesses smooth cell decomposition. We prove that for every definable open set, the definable indefinitely continuously differentiable functions are a dense subset of the definable continuous function with respect to the $ o$-minimal Whitney topology.

References [Enhancements On Off] (What's this?)

  • 1. van den Dries, L., Tame topology and o-minimal structures. London Mathematical Society Lecture Note Series, 248. Cambridge University Press, Cambridge, 1998. MR 1633348 (99j:03001)
  • 2. van den Dries, L., and Miller, C., Geometric categories and o-minimal structures. Duke Math. J. 84 (1996), no. 2, 497-540. MR 1404337 (97i:32008)
  • 3. Efroymson, G. A., The extension theorem for Nash functions. Real algebraic geometry and quadratic forms (Rennes, 1981), pp. 343-357, Lecture Notes in Math., 959, Springer, Berlin-New York, 1982. MR 683141 (84i:58002)
  • 4. Escribano, J., Approximation theorems in o-minimal structures, Illinois Journal of Mathematics 46(1) (2002), 111-128. MR 1936078 (2003i:03042)
  • 5. Jones, G. O., Local to global methods in o-minimal expansions of fields. Doctoral Thesis, Wolfson College University of Oxford, 2006.
  • 6. Pecker, D., On Efroymson's extension theorem for Nash functions. J. Pure Appl. Algebra 37 (1985), no. 2, 193-203. MR 796409 (87e:58002)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 03C64, 26E10

Retrieve articles in all journals with MSC (2000): 03C64, 26E10

Additional Information

Andreas Fischer
Affiliation: Department of Mathematics and Statistics, University of Saskatchewan, 106 Wiggins Road, Saskatoon, SK, S7N 5E6, Canada

Keywords: $o$-minimal structures, exponential function, approximation
Received by editor(s): January 31, 2007
Received by editor(s) in revised form: April 10, 2007, and May 15, 2007
Published electronically: February 29, 2008
Additional Notes: This research was partially supported by the NSERC discovery grant of Dr. Salma Kuhlmann
Communicated by: Julia Knight
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society