Restrictions of smooth functions to a closed subset
[Restrictions de fonctions différentiables à un sous ensemble fermé]
Annales de l'Institut Fourier, Tome 54 (2004) no. 6, pp. 1811-1826.

Nous proposons une approche d’une conjecture de Bierstone-Milman-Pawłucki sur le problème de Whitney concernant le prolongement C d des fonctions. Elle permet de montrer que la conjecture est vraie pour des ensembles fractals classiques. Nous obtenons ensuite un raffinement d’un théorème de Spallek sur la platitude.

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.

DOI : 10.5802/aif.2067
Classification : 26B05
Keywords: Whitney's problem, Spallek's theorem, smooth functions, higher order paratangent bundle, flatness, multi-dimensional Vandermonde matrix, self-similar set
Mot clés : Problème de Whitney, théorème de Spallek, fonction différentiable, fibré paratangent d'ordre supérieur, platitude, matrice de Vandermonde multi-dimensionnelle
Izumi, Shuzo 1

1 Kinki University,Department of Mathematics, Kowakae Higashi-Osaka 577-8502 (Japan)
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Izumi, Shuzo. Restrictions of smooth functions to a closed subset. Annales de l'Institut Fourier, Tome 54 (2004) no. 6, pp. 1811-1826. doi : 10.5802/aif.2067. http://www.numdam.org/articles/10.5802/aif.2067/

[AS] A.A. Akopyan; A.A. Saakyan Multivariate splines and polynomial interpolation, Russian Math. Surveys, Volume 48 (1993) no. 5, pp. 1-72 | MR | Zbl

[B] G. Bouligand Introduction à la géométrie infinitésimale directe, Vuibert, Paris, 1932 | JFM | Zbl

[BMP1] E. Bierstone; P. Milman; W. Pawłucki Composite differentiable functions, Duke Math. J., Volume 83 (1996), pp. 607-620 | Zbl

[BMP2] E. Bierstone; P. Milman; W. Pawłucki Differentiable functions defined in closed sets. A problem of Whitney, Invent. Math., Volume 151 (2003), pp. 329-352 | Zbl

[C] G. Choquet Convergence, Ann. Inst. Fourier, Volume 23 (1948), pp. 57-112 | Numdam | Zbl

[F] K. Falconer Techniques in fractal geometry, John-Wiley and Sons, 1997 | MR | Zbl

[G1] G. Glaeser Études de quelques algèbres tayloriennes, J. Anal. Math., Volume 6 (1958), pp. 1-124 | MR | Zbl

[G2] G. Glaeser; ed. C.T.C. Wall L'interpolation des fonctions différentiables de plusieurs variables, Proceedings of Liverpool singularities symposium II (Lecture Notes in Math.), Volume 209 (1971), pp. 1-33 | Zbl

[I] S. Izumi Flatness of differentiable functions along a subset of a real analytic set, J. Anal. Math., Volume 86 (2002), pp. 235-246 | MR | Zbl

[K] P. Kergin A natural interpolation of 𝒞 K functions, J. Approx. Theory, Volume 29 (1980), pp. 278-293 | MR | Zbl

[MM] C.A. Micchelli; P. Milman A formula for Kergin interpolation in k , J. Approx. Theory, Volume 29 (1980), pp. 294-296 | MR | Zbl

[S] K. Spallek -Platte Funktionen auf semianalytischen Mengen, Math. Ann., Volume 227 (1977), pp. 277-286 | MR | Zbl

[W1] H. Whitney Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., Volume 36 (1934), pp. 63-89 | JFM | MR | Zbl

[W2] H. Whitney Differentiable functions defined in closed sets. I, Trans. Amer. Math. Soc., Volume 36 (1934) no. 2, pp. 369-387 | MR | Zbl

[YHK] M. Yamaguti; M. Hata; J. Kigami Mathematics of Fractals, Transl. Math. Monog., 167, Amer. Math. Soc., Providence, 1997 | MR | Zbl

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