Restrictions of smooth functions to a closed subset

Shuzo Izumi[1]

  • [1] Kinki University,Department of Mathematics, Kowakae Higashi-Osaka 577-8502 (Japan)

Annales de l’institut Fourier (2004)

  • Volume: 54, Issue: 6, page 1811-1826
  • ISSN: 0373-0956

Abstract

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We first provide an approach to the conjecture of Bierstone-Milman-Pawłucki on Whitney’s problem on C d extendability of functions. For example, the conjecture is affirmative for classical fractal sets. Next, we give a sharpened form of Spallek’s theorem on flatness.

How to cite

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Izumi, Shuzo. "Restrictions of smooth functions to a closed subset." Annales de l’institut Fourier 54.6 (2004): 1811-1826. <http://eudml.org/doc/116160>.

@article{Izumi2004,
abstract = {We first provide an approach to the conjecture of Bierstone-Milman-Pawłucki on Whitney’s problem on $C^d$ extendability of functions. For example, the conjecture is affirmative for classical fractal sets. Next, we give a sharpened form of Spallek’s theorem on flatness.},
affiliation = {Kinki University,Department of Mathematics, Kowakae Higashi-Osaka 577-8502 (Japan)},
author = {Izumi, Shuzo},
journal = {Annales de l’institut Fourier},
keywords = {Whitney's problem; Spallek's theorem; smooth functions; higher order paratangent bundle; flatness; multi-dimensional Vandermonde matrix; self-similar set; Whitney problem; Spallek theorem},
language = {eng},
number = {6},
pages = {1811-1826},
publisher = {Association des Annales de l'Institut Fourier},
title = {Restrictions of smooth functions to a closed subset},
url = {http://eudml.org/doc/116160},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Izumi, Shuzo
TI - Restrictions of smooth functions to a closed subset
JO - Annales de l’institut Fourier
PY - 2004
PB - Association des Annales de l'Institut Fourier
VL - 54
IS - 6
SP - 1811
EP - 1826
AB - We first provide an approach to the conjecture of Bierstone-Milman-Pawłucki on Whitney’s problem on $C^d$ extendability of functions. For example, the conjecture is affirmative for classical fractal sets. Next, we give a sharpened form of Spallek’s theorem on flatness.
LA - eng
KW - Whitney's problem; Spallek's theorem; smooth functions; higher order paratangent bundle; flatness; multi-dimensional Vandermonde matrix; self-similar set; Whitney problem; Spallek theorem
UR - http://eudml.org/doc/116160
ER -

References

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  2. E. Bierstone, P. Milman, W. Pawłucki, Composite differentiable functions, Duke Math. J. 83 (1996), 607-620 Zbl0868.32011
  3. E. Bierstone, P. Milman, W. Pawłucki, Differentiable functions defined in closed sets. A problem of Whitney, Invent. Math. 151 (2003), 329-352 Zbl1031.58002
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  9. S. Izumi, Flatness of differentiable functions along a subset of a real analytic set, J. Anal. Math. 86 (2002), 235-246 Zbl1032.32007MR1894483
  10. P. Kergin, A natural interpolation of K functions, J. Approx. Theory 29 (1980), 278-293 Zbl0492.41008MR598722
  11. C.A. Micchelli, P. Milman, A formula for Kergin interpolation in ℝ k , J. Approx. Theory 29 (1980), 294-296 Zbl0454.41002MR598723
  12. K. Spallek, ℓ -Platte Funktionen auf semianalytischen Mengen, Math. Ann. 227 (1977), 277-286 Zbl0333.32025MR450630
  13. H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), 63-89 Zbl0008.24902MR1501735
  14. H. Whitney, Differentiable functions defined in closed sets. I, Trans. Amer. Math. Soc. 36 (1934), 369-387 Zbl0009.20803MR1501749
  15. M. Yamaguti, M. Hata, J. Kigami, Mathematics of Fractals, 167 (1997), Amer. Math. Soc., Providence Zbl0888.58030MR1471705

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