Examples of functions -extendable for each finite, but not ∞ -extendable

Wiesław Pawłucki

Banach Center Publications (1998)

  • Volume: 44, Issue: 1, page 183-187
  • ISSN: 0137-6934

Abstract

top
In Example 1, we describe a subset X of the plane and a function on X which has a k -extension to the whole ℝ 2 for each finite, but has no ∞ -extension to ℝ 2 . In Example 2, we construct a similar example of a subanalytic subset of ℝ 5 ; much more sophisticated than the first one. The dimensions given here are smallest possible.

How to cite

top

Pawłucki, Wiesław. "Examples of functions $^$-extendable for each finite, but not $^∞$-extendable." Banach Center Publications 44.1 (1998): 183-187. <http://eudml.org/doc/208881>.

@article{Pawłucki1998,
abstract = {In Example 1, we describe a subset X of the plane and a function on X which has a $^k$-extension to the whole $ℝ^2$ for each finite, but has no $^∞$-extension to $ℝ^2$. In Example 2, we construct a similar example of a subanalytic subset of $ℝ^5$; much more sophisticated than the first one. The dimensions given here are smallest possible.},
author = {Pawłucki, Wiesław},
journal = {Banach Center Publications},
keywords = {extension of -functions; subanalytic sets; algebras of germs of analytic functions},
language = {eng},
number = {1},
pages = {183-187},
title = {Examples of functions $^$-extendable for each finite, but not $^∞$-extendable},
url = {http://eudml.org/doc/208881},
volume = {44},
year = {1998},
}

TY - JOUR
AU - Pawłucki, Wiesław
TI - Examples of functions $^$-extendable for each finite, but not $^∞$-extendable
JO - Banach Center Publications
PY - 1998
VL - 44
IS - 1
SP - 183
EP - 187
AB - In Example 1, we describe a subset X of the plane and a function on X which has a $^k$-extension to the whole $ℝ^2$ for each finite, but has no $^∞$-extension to $ℝ^2$. In Example 2, we construct a similar example of a subanalytic subset of $ℝ^5$; much more sophisticated than the first one. The dimensions given here are smallest possible.
LA - eng
KW - extension of -functions; subanalytic sets; algebras of germs of analytic functions
UR - http://eudml.org/doc/208881
ER -

References

top
  1. [1] E. Bierstone, P. D. Milman, Geometric and differential properties of subanalytic sets, Bull. Amer. Math. Soc. 25 (1991), 385-383. Zbl0739.32010
  2. [2] E. Bierstone, P. D. Milman, Geometric and differential properties of subanalytic sets, preprint. Zbl0912.32006
  3. [3] E. Bierstone, P. D. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 5-42. Zbl0674.32002
  4. [4] E. Bierstone, P. D. Milman, W. Pawłucki, Composite differentiable functions, Duke Math. J. 83 (1996), 607-620. Zbl0868.32011
  5. [5] Z. Denkowska, S. Łojasiewicz, J. Stasica, Certaines propriétés élémentaires des ensembles sous-analytiques, Bull. Polish Acad. Sci. Math. 27 (1979), 529-536. Zbl0435.32006
  6. [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. 
  7. [7] H. Hironaka, Subanalytic sets, in: Number Theory, Algebraic Geometry and Commutative Algebra in Honor of Yasuo Akizuki, Kinokuniya, Tokyo, 1973, 453-493. 
  8. [8] B. Malgrange, Ideals of Differentiable Functions, Oxford University Press, Bombay, 1966. 
  9. [9] J. Merrien, Prolongateurs de fonctions différentiables d'une variable réelle, J. Math. Pures Appl. (9) 45 (1966), 291-309. Zbl0163.06602
  10. [10] W. Pawłucki, On relations among analytic functions and geometry of subanalytic sets, Bull. Polish Acad. Sci. Math. 37 (1989), 117-125. Zbl0769.32003
  11. [11] W. Pawłucki, On Gabrielov's regularity condition for analytic mappings, Duke Math. J. 65 (1992), 299-311. Zbl0773.32009

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.