Smooth approximations without critical points

Petr Hájek; Michal Johanis

Open Mathematics (2003)

  • Volume: 1, Issue: 3, page 284-291
  • ISSN: 2391-5455

Abstract

top
In any separable Banach space containing c 0 which admits a C k-smooth bump, every continuous function can be approximated by a C k-smooth function whose range of derivative is of the first category. Moreover, the approximation can be constructed in such a way that its derivative avoids a prescribed countable set (in particular the approximation can have no critical points). On the other hand, in a Banach space with the RNP, the range of the derivative of every smooth bounded bump contains a set residual in some neighbourhood of zero.

How to cite

top

Petr Hájek, and Michal Johanis. "Smooth approximations without critical points." Open Mathematics 1.3 (2003): 284-291. <http://eudml.org/doc/268737>.

@article{PetrHájek2003,
abstract = {In any separable Banach space containing c 0 which admits a C k-smooth bump, every continuous function can be approximated by a C k-smooth function whose range of derivative is of the first category. Moreover, the approximation can be constructed in such a way that its derivative avoids a prescribed countable set (in particular the approximation can have no critical points). On the other hand, in a Banach space with the RNP, the range of the derivative of every smooth bounded bump contains a set residual in some neighbourhood of zero.},
author = {Petr Hájek, Michal Johanis},
journal = {Open Mathematics},
keywords = {46B20; 46G05},
language = {eng},
number = {3},
pages = {284-291},
title = {Smooth approximations without critical points},
url = {http://eudml.org/doc/268737},
volume = {1},
year = {2003},
}

TY - JOUR
AU - Petr Hájek
AU - Michal Johanis
TI - Smooth approximations without critical points
JO - Open Mathematics
PY - 2003
VL - 1
IS - 3
SP - 284
EP - 291
AB - In any separable Banach space containing c 0 which admits a C k-smooth bump, every continuous function can be approximated by a C k-smooth function whose range of derivative is of the first category. Moreover, the approximation can be constructed in such a way that its derivative avoids a prescribed countable set (in particular the approximation can have no critical points). On the other hand, in a Banach space with the RNP, the range of the derivative of every smooth bounded bump contains a set residual in some neighbourhood of zero.
LA - eng
KW - 46B20; 46G05
UR - http://eudml.org/doc/268737
ER -

References

top
  1. [1] D. Azagra and M. Cepedello Boiso: “Uniform approximation of continuous mappings by smooth mappings with no critical points on Hilbert manifolds”, preprint, http://arxiv.org/archive/math. Zbl1060.57015
  2. [2] D. Azagra and R. Deville: “James’ theorem fails for starlike bodies”, J. Funct. Anal., Vol. 180, (2001), pp. 328–346. http://dx.doi.org/10.1006/jfan.2000.3696 Zbl0983.46016
  3. [3] D. Azagra, R. Deville, M. Jiménez-Sevilla: “On the range of the derivatives of a smooth mapping between Banach spaces”, to appear in Proc. Cambridge Phil. Soc. Zbl1034.46039
  4. [4] D. Azagra, M. Fabian, M. Jiménez-Sevilla: “Exact filling in figures by the derivatives of smooth mappings between Banach spaces”, preprint. Zbl1109.46041
  5. [5] D. Azagra and M. Jiménez-Sevilla: “The failure of Rolle’s Theorem in infinite dimensional Banach spaces”, J. Funct. Anal., Vol. 182, (2001), pp. 207–226. http://dx.doi.org/10.1006/jfan.2000.3709 Zbl0995.46025
  6. [6] D. Azagra and M. Jiménez-Sevilla: “Geometrical and topological properties of starlike bodies and bumps in Banach spaces”, to appear in Extracta Math. Zbl1028.46025
  7. [7] J.M. Borwein, M. Fabian, I. Kortezov, P.D. Loewen: “The range of the gradient of a continuously differentiable bump”, J. Nonlinear and Convex Anal., Vol. 2, (2001), pp. 1–19. Zbl0993.46023
  8. [8] J.M. Borwein, M. Fabian, P.D. Loewen: “The range of the gradient of a Lipschiz C 1-smooth bump in infinite dimensions”, to appear in Israel Journal of Mathematics. Zbl1010.46016
  9. [9] R. Deville, G. Godefroy, V. Zizler: “Smoothness and renormings in Banach spaces”, Monographs and Surveys in Pure and Applied Mathematics 64, Pitman, 1993. Zbl0782.46019
  10. [10] M. Fabian, O. Kalenda, J. Kolář: “Filling analytic sets by the derivatives of C 1-smooth bumps”, to appear in Proc. Amer. Math. Soc. Zbl1107.46032
  11. [11] M. Fabian, J.H.M. Whitfield, V. Zizler: “Norms with locally Lipschizian derivatives”, Israel Journal of Mathematics, Vol. 44, (1983), pp. 262–276. Zbl0521.46009
  12. [12] T. Gaspari: “On the range of the derivative of a real valued function with bounded support”, preprint. Zbl1033.46036
  13. [13] P. Hájek: “Smooth functions on c 0”, Israel Journal of Mathematics, Vol. 104, (1998), pp. 17–27. Zbl0940.46023
  14. [14] A. Sobczyk: “Projection of the space m on its subspace c 0”, Bull. Amer. Math. Soc., Vol. 47, (1941), pp. 938–947. http://dx.doi.org/10.1090/S0002-9904-1941-07593-2 Zbl0027.40801
  15. [15] C. Stegall: “Optimization of functions on certain subsets of Banach spaces”, Math. Ann., Vol. 236, (1978), pp. 171–176. http://dx.doi.org/10.1007/BF01351389 Zbl0365.49006

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.