Geodesic
Extending Peano derivatives
Mathematica Bohemica, Tome 119 (1994) no. 4, pp. 387-406

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

MR Zbl
Let $H\subset [0,1]$ be a closed set, $k$ a positive integer and $f$ a function defined on $H$ so that the $k$-th Peano derivative relative to $H$ exists. The major result of this paper is that if $H$ has finite Denjoy index, then $f$ has an extension, $F$, to $[0,1]$ which is $k$ times Peano differentiable on $[0,1]$ with $f_i=F_i$ on $H$ for $i=1,2,\ldots,k$.
Let $H\subset [0,1]$ be a closed set, $k$ a positive integer and $f$ a function defined on $H$ so that the $k$-th Peano derivative relative to $H$ exists. The major result of this paper is that if $H$ has finite Denjoy index, then $f$ has an extension, $F$, to $[0,1]$ which is $k$ times Peano differentiable on $[0,1]$ with $f_i=F_i$ on $H$ for $i=1,2,\ldots,k$.
DOI : 10.21136/MB.1994.126113
Classification : 26A24
Keywords: Peano derivatives; Denjoy index 
Fejzić, Hajrudin; Mařík, Jan; Weil, Clifford E. Extending Peano derivatives. Mathematica Bohemica, Tome 119 (1994) no. 4, pp. 387-406. doi: 10.21136/MB.1994.126113
@article{10_21136_MB_1994_126113,
     author = {Fejzi\'c, Hajrudin and Ma\v{r}{\'\i}k, Jan and Weil, Clifford E.},
     title = {Extending {Peano} derivatives},
     journal = {Mathematica Bohemica},
     pages = {387--406},
     year = {1994},
     volume = {119},
     number = {4},
     doi = {10.21136/MB.1994.126113},
     mrnumber = {1316592},
     zbl = {0824.26003},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.1994.126113/}
}
TY  - JOUR
AU  - Fejzić, Hajrudin
AU  - Mařík, Jan
AU  - Weil, Clifford E.
TI  - Extending Peano derivatives
JO  - Mathematica Bohemica
PY  - 1994
SP  - 387
EP  - 406
VL  - 119
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.1994.126113/
DO  - 10.21136/MB.1994.126113
LA  - en
ID  - 10_21136_MB_1994_126113
ER  - 
%0 Journal Article
%A Fejzić, Hajrudin
%A Mařík, Jan
%A Weil, Clifford E.
%T Extending Peano derivatives
%J Mathematica Bohemica
%D 1994
%P 387-406
%V 119
%N 4
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.1994.126113/
%R 10.21136/MB.1994.126113
%G en
%F 10_21136_MB_1994_126113

[1] Z. Buczolich: Second Peano derivatives are not extendable. Real Analysis Exch 14 (1988-89), 423-428. | DOI | MR

[2] P. Bullen: Denjoy's index and porosity. Real Analysis Exch, 10 (1984-84), 85-144. | DOI | MR

[3] A. Denjoy: Sur l'integration des coefficients differentiels d'order supérieur. Fundamenta Mathematicae 25 (1935), 273-326. | DOI

[4] M. J. Evans C. E. Weil: Peano derivatives: A survey. Real Analysis Exch, 7(1981-82), 5-24. | MR

[5] H. Fejzić: Decomposition of Peano derivatives. Proc. Amer. Soc 119 (1993), no. 2, 599-609. | DOI | MR

[6] H. Fejzić: The Peano derivatives. Doct. Dissertation. Michigan State University, 1992.

[7] H. Fejzić: On generalized Peano and Peano derivatives. Fundamenta Mathematicae 143 (1994), 55-74. | DOI | MR

[8] V. Jarník: Sur l'extension du domaine de definition des fonctions d'une variable, qui laisse intacte la derivabilite de la fonction. Bull international de l'Acad Sci de Boheme (1923).

[9] J. Mařík: Derivatives and closed sets. Acta Math. Hung. 43 (1-2) (1984), 25-29. | MR

[10] G. Petruska, M. Laczkovich: Baire 1 functions, approximately continuous functions and derivatives. Acta Math Acad Sci Hungar, 25 (1974), 189-212. | DOI | MR | Zbl

[11] C. E. Weil: The Peano derivative: What's known and what isn't. Real Analysis Exchange 9 (1983-1984), 354-365. | DOI | MR

Cité par Sources :