Extending Peano derivatives: necessary and sufficient conditions

• Autorzy: Hans Volkmer
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1999
• Tom: 159
• Numer: 3
• Strony: 219-229

Daty

wydano

1999

otrzymano

1997-06-18

poprawiono

1998-02-02

poprawiono

1998-09-24

Twórcy

autor

Hans Volkmer

• Department of Mathematical Sciences, University of Wisconsin-Milwaukee, P.O. Box 413, Milwaukee, Wisconsin 53201 U.S.A.,

Bibliografia

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