Extending Peano derivatives: necessary and sufficient conditions

Hans Volkmer

Fundamenta Mathematicae (1999)

  • Volume: 159, Issue: 3, page 219-229
  • ISSN: 0016-2736

Abstract

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The paper treats functions which are defined on closed subsets of [0,1] and which are k times Peano differentiable. A necessary and sufficient condition is given for the existence of a k times Peano differentiable extension of such a function to [0,1]. Several applications of the result are presented. In particular, functions defined on symmetric perfect sets are studied.

How to cite

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Volkmer, Hans. "Extending Peano derivatives: necessary and sufficient conditions." Fundamenta Mathematicae 159.3 (1999): 219-229. <http://eudml.org/doc/212330>.

@article{Volkmer1999,
abstract = {The paper treats functions which are defined on closed subsets of [0,1] and which are k times Peano differentiable. A necessary and sufficient condition is given for the existence of a k times Peano differentiable extension of such a function to [0,1]. Several applications of the result are presented. In particular, functions defined on symmetric perfect sets are studied.},
author = {Volkmer, Hans},
journal = {Fundamenta Mathematicae},
keywords = {Denjoy index; Peano derivatives; symmetric perfect sets; extension},
language = {eng},
number = {3},
pages = {219-229},
title = {Extending Peano derivatives: necessary and sufficient conditions},
url = {http://eudml.org/doc/212330},
volume = {159},
year = {1999},
}

TY - JOUR
AU - Volkmer, Hans
TI - Extending Peano derivatives: necessary and sufficient conditions
JO - Fundamenta Mathematicae
PY - 1999
VL - 159
IS - 3
SP - 219
EP - 229
AB - The paper treats functions which are defined on closed subsets of [0,1] and which are k times Peano differentiable. A necessary and sufficient condition is given for the existence of a k times Peano differentiable extension of such a function to [0,1]. Several applications of the result are presented. In particular, functions defined on symmetric perfect sets are studied.
LA - eng
KW - Denjoy index; Peano derivatives; symmetric perfect sets; extension
UR - http://eudml.org/doc/212330
ER -

References

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  1. [1] Z. Buczolich, Second Peano derivatives are not extendable, Real Anal. Exchange 14 (1988-89), 423-428. Zbl0679.26005
  2. [2] Z. Buczolich and C. Weil, Extending Peano differentiable functions, Atti Sem. Mat. Fis. Univ. Modena 44 (1996), 323-330. Zbl0865.26007
  3. [3] P. Bullen, Denjoy's index and porosity, Real Anal. Exchange 10 (1984-85), 85-144. Zbl0595.26001
  4. [4] A. Denjoy, Sur l'intégration des coefficients différentiels d'ordre supérieur, Fund. Math. 25 (1935), 273-326. Zbl61.1115.03
  5. [5] A. Denjoy, Leçons sur le calcul de coefficients d'une série trigonométrique I-IV, Gauthier-Villars, Paris, 1941-1949. 
  6. [6] H. Fejzić, The Peano derivatives, doct. dissertation, Michigan State Univ., 1992. 
  7. [7] H. Fejzić, J. Mařík and C. Weil, Extending Peano derivatives, Math. Bohem. 119 (1994), 387-406. Zbl0824.26003
  8. [8] H. W. Oliver, The exact Peano derivative, Trans. Amer. Math. Soc. 76 (1954), 444-456. Zbl0055.28505
  9. [9] B. Thomson, Real Functions, Lecture Notes in Math. 1170, Springer, Berlin, 1985. 

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