Examples of functions -extendable for each finite, but not ∞-extendable
Pawłucki, Wiesław
Banach Center Publications, Tome 43 (1998), p. 183-187 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

In Example 1, we describe a subset X of the plane and a function on X which has a k-extension to the whole ℝ2 for each finite, but has no ∞-extension to ℝ2. In Example 2, we construct a similar example of a subanalytic subset of ℝ5; much more sophisticated than the first one. The dimensions given here are smallest possible.

Publié le : 1998-01-01
EUDML-ID : urn:eudml:doc:208881 
@article{bwmeta1.element.bwnjournal-article-bcpv44i1p183bwm,
     author = {Paw\l ucki, Wies\l aw},
     title = {Examples of functions $^$-extendable for each finite, but not $^$\infty$$-extendable},
     journal = {Banach Center Publications},
     volume = {43},
     year = {1998},
     pages = {183-187},
     zbl = {0945.32002},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv44i1p183bwm}
}

Pawłucki, Wiesław. Examples of functions $^$-extendable for each finite, but not $^∞$-extendable. Banach Center Publications, Tome 43 (1998) pp. 183-187. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv44i1p183bwm/

[000] [1] E. Bierstone, P. D. Milman, Geometric and differential properties of subanalytic sets, Bull. Amer. Math. Soc. 25 (1991), 385-383. | Zbl 0739.32010

[001] [2] E. Bierstone, P. D. Milman, Geometric and differential properties of subanalytic sets, preprint. | Zbl 0912.32006

[002] [3] E. Bierstone, P. D. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 5-42. | Zbl 0674.32002

[003] [4] E. Bierstone, P. D. Milman, W. Pawłucki, Composite differentiable functions, Duke Math. J. 83 (1996), 607-620. | Zbl 0868.32011

[004] [5] Z. Denkowska, S. Łojasiewicz, J. Stasica, Certaines propriétés élémentaires des ensembles sous-analytiques, Bull. Polish Acad. Sci. Math. 27 (1979), 529-536. | Zbl 0435.32006

[005] [6] A. Gabrielov, Projections of semianalytic sets, Funkcional Anal. i Priložen. 2 no. 4 (1968), 18-30 (in Russian). English transl.: Functional Anal. Appl. 2 (1968), 282-291.

[006] [7] H. Hironaka, Subanalytic sets, in: Number Theory, Algebraic Geometry and Commutative Algebra in Honor of Yasuo Akizuki, Kinokuniya, Tokyo, 1973, 453-493.

[007] [8] B. Malgrange, Ideals of Differentiable Functions, Oxford University Press, Bombay, 1966.

[008] [9] J. Merrien, Prolongateurs de fonctions différentiables d'une variable réelle, J. Math. Pures Appl. (9) 45 (1966), 291-309. | Zbl 0163.06602

[009] [10] W. Pawłucki, On relations among analytic functions and geometry of subanalytic sets, Bull. Polish Acad. Sci. Math. 37 (1989), 117-125. | Zbl 0769.32003

[010] [11] W. Pawłucki, On Gabrielov's regularity condition for analytic mappings, Duke Math. J. 65 (1992), 299-311. | Zbl 0773.32009