Let Ω,F,G be a partition of such that Ω is open, F is and of the first category, and G is . 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.
@article{bwmeta1.element.bwnjournal-article-fmv151i2p149bwm,
author = {Jean Schmets and Manuel Valdivia},
title = {The Zahorski theorem is valid in Gevrey classes},
journal = {Fundamenta Mathematicae},
volume = {149},
year = {1996},
pages = {149-166},
zbl = {0877.26014},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv151i2p149bwm}
}
Schmets, Jean; Valdivia, Manuel. The Zahorski theorem is valid in Gevrey classes. Fundamenta Mathematicae, Tome 149 (1996) pp. 149-166. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv151i2p149bwm/
[00000] [1] R. P. Boas, A theorem on analytic functions of a real variable, Bull. Amer. Math. Soc. 41 233-236(1935). | Zbl 0011.20105
[00001] [2] S. Mandelbrojt, Séries entières et transformées de Fourier, Applications, Publ. Math. Soc. Japan 10, 1967. | Zbl 0172.09302
[00002] [3] H. Salzmann and K. Zeller, Singularitäten unendlich oft differenzierbarer Funktionen, Math. Z. 62 354-367 (1955). | Zbl 0064.29903
[00003] [4] J. Siciak, Punkty regularne i osobliwe funkcji klasy [Regular and singular points of functions], Zeszyty Nauk. Polit. Śląsk. Ser. Mat.-Fiz. 48 (853) (1986), 127-146 (in Polish).
[00004] [5] H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 63-89 (1934). | Zbl 0008.24902
[00005] [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). | Zbl 0033.25504