The Zahorski theorem is valid in Gevrey classes

Jean Schmets; Manuel Valdivia

Fundamenta Mathematicae (1996)

  • Volume: 151, Issue: 2, page 149-166
  • ISSN: 0016-2736

Abstract

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Let Ω,F,G be a partition of ℝ n such that Ω is open, F is F σ and of the first category, and G is G δ . We prove that, for every γ ∈ ]1,∞[, there is an element of the Gevrey class Γγ which is analytic on Ω, has F as its set of defect points and has G as its set of divergence points.

How to cite

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Schmets, Jean, and Valdivia, Manuel. "The Zahorski theorem is valid in Gevrey classes." Fundamenta Mathematicae 151.2 (1996): 149-166. <http://eudml.org/doc/212187>.

@article{Schmets1996,
abstract = {Let Ω,F,G be a partition of $ℝ^n$ such that Ω is open, F is $F_σ$ and of the first category, and G is $G_δ$. We prove that, for every γ ∈ ]1,∞[, there is an element of the Gevrey class Γγ which is analytic on Ω, has F as its set of defect points and has G as its set of divergence points.},
author = {Schmets, Jean, Valdivia, Manuel},
journal = {Fundamenta Mathematicae},
keywords = {Gevrey classes; defect point; divergence point; -functions; Gevrey functions; multivariate Zahorski theorem; Gevrey class},
language = {eng},
number = {2},
pages = {149-166},
title = {The Zahorski theorem is valid in Gevrey classes},
url = {http://eudml.org/doc/212187},
volume = {151},
year = {1996},
}

TY - JOUR
AU - Schmets, Jean
AU - Valdivia, Manuel
TI - The Zahorski theorem is valid in Gevrey classes
JO - Fundamenta Mathematicae
PY - 1996
VL - 151
IS - 2
SP - 149
EP - 166
AB - Let Ω,F,G be a partition of $ℝ^n$ such that Ω is open, F is $F_σ$ and of the first category, and G is $G_δ$. We prove that, for every γ ∈ ]1,∞[, there is an element of the Gevrey class Γγ which is analytic on Ω, has F as its set of defect points and has G as its set of divergence points.
LA - eng
KW - Gevrey classes; defect point; divergence point; -functions; Gevrey functions; multivariate Zahorski theorem; Gevrey class
UR - http://eudml.org/doc/212187
ER -

References

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  1. [1] R. P. Boas, A theorem on analytic functions of a real variable, Bull. Amer. Math. Soc. 41 233-236(1935). Zbl0011.20105
  2. [2] S. Mandelbrojt, Séries entières et transformées de Fourier, Applications, Publ. Math. Soc. Japan 10, 1967. Zbl0172.09302
  3. [3] H. Salzmann and K. Zeller, Singularitäten unendlich oft differenzierbarer Funktionen, Math. Z. 62 354-367 (1955). Zbl0064.29903
  4. [4] J. Siciak, Punkty regularne i osobliwe funkcji klasy C ∞ [Regular and singular points of C ∞ functions], Zeszyty Nauk. Polit. Śląsk. Ser. Mat.-Fiz. 48 (853) (1986), 127-146 (in Polish). 
  5. [5] H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 63-89 (1934). Zbl0008.24902
  6. [6] Z. Zahorski, Sur l'ensemble des points singuliers d'une fonction d'une variable réelle admettant les dérivées de tous les ordres, Fund. Math. 34 183-245(1947). Zbl0033.25504

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