# On generalized Peano and Peano derivatives

Fundamenta Mathematicae (1993)

- Volume: 143, Issue: 1, page 55-74
- ISSN: 0016-2736

## Access Full Article

top## Abstract

top## How to cite

topFejzić, H.. "On generalized Peano and Peano derivatives." Fundamenta Mathematicae 143.1 (1993): 55-74. <http://eudml.org/doc/211992>.

@article{Fejzić1993,

abstract = {A function F is said to have a generalized Peano derivative at x if F is continuous in a neighborhood of x and if there exists a positive integer q such that a qth primitive of F in the neighborhood has the (q+n)th Peano derivative at x; in this case the latter is called the generalized nth Peano derivative of F at x and denoted by $F_\{[n]\}(x)$. We show that generalized Peano derivatives belong to the class [Δ’]. Also we show that they are path derivatives with a nonporous system of paths satisfying the I.I.C. condition as defined in [3]. This gives a new approach to studying generalized Peano and Peano derivatives since all their known properties can be obtained from the corresponding properties of path derivatives. Moreover, generalized Peano derivatives are selective derivatives.},

author = {Fejzić, H.},

journal = {Fundamenta Mathematicae},

keywords = {composite derivatives; generalized Peano derivative; compact derivative; path derivative; selective derivative},

language = {eng},

number = {1},

pages = {55-74},

title = {On generalized Peano and Peano derivatives},

url = {http://eudml.org/doc/211992},

volume = {143},

year = {1993},

}

TY - JOUR

AU - Fejzić, H.

TI - On generalized Peano and Peano derivatives

JO - Fundamenta Mathematicae

PY - 1993

VL - 143

IS - 1

SP - 55

EP - 74

AB - A function F is said to have a generalized Peano derivative at x if F is continuous in a neighborhood of x and if there exists a positive integer q such that a qth primitive of F in the neighborhood has the (q+n)th Peano derivative at x; in this case the latter is called the generalized nth Peano derivative of F at x and denoted by $F_{[n]}(x)$. We show that generalized Peano derivatives belong to the class [Δ’]. Also we show that they are path derivatives with a nonporous system of paths satisfying the I.I.C. condition as defined in [3]. This gives a new approach to studying generalized Peano and Peano derivatives since all their known properties can be obtained from the corresponding properties of path derivatives. Moreover, generalized Peano derivatives are selective derivatives.

LA - eng

KW - composite derivatives; generalized Peano derivative; compact derivative; path derivative; selective derivative

UR - http://eudml.org/doc/211992

ER -

## References

top- [1] S. Agronsky, R. Biskner, A. Bruckner and J. Mařík, Representations of functions by derivatives, Trans. Amer. Math. Soc. 263 (1981), 493-500. Zbl0455.26002
- [2] A. M. Bruckner, Differentiation of Real Functions, Lecture Notes in Math. 659, Springer, Berlin 1978. Zbl0382.26002
- [3] A.M. Bruckner, R.J. O'Malley and B.S. Thomson, Path derivatives: A unified view of certain generalized derivatives, Trans. Amer. Math. Soc. 283 (1984), 97-125. Zbl0541.26003
- [4] M. E. Corominas, Contribution à la théorie de la dérivation d'ordre supérieur, Bull. Soc. Math. France 81 (1953), 176-222.
- [5] A. Denjoy, Sur l'intégration des coefficients différentiels d'ordre supérieur, Fund. Math. 25 (1935), 273-326. Zbl61.1115.03
- [6] H. Fejzić, Decomposition of Peano derivatives, Proc. Amer. Math. Soc., to appear. Zbl0797.26003
- [7] M. Laczkovich, On the absolute Peano derivatives, Ann. Univ. Sci. Budapest. Eőtvős Sect. Math. 21 (1978), 83-97. Zbl0425.26005
- [8] C. M. Lee, On absolute Peano derivatives, Real Anal. Exchange 8 (1982-1983), 228-243.
- [9] C. M. Lee, On generalized Peano derivatives, Trans. Amer. Math. Soc. 275 (1983), 381-396. Zbl0506.26006
- [10] H. Oliver, The exact Peano derivative, ibid. 76 (1954), 444-456. Zbl0055.28505
- [11] R. J. O'Malley, Decomposition of approximate derivatives, Proc. Amer. Math. Soc. 69 (1978), 243-247.
- [12] R. J. O'Malley and C. E. Weil, The oscillatory behavior of certain derivatives, Trans. Amer. Math. Soc. 234 (1977), 467-481. Zbl0372.26006
- [13] J. Mařík, On generalized derivatives, Real Anal. Exchange 3 (1977-78), 87-92.
- [14] J. Mařík, Derivatives and closed sets, Acta Math. Hungar. 43 (1-2) (1984), 25-29. Zbl0543.26003
- [15] S. Verblunsky, On the Peano derivatives, Proc. London Math. Soc. (3) 22 (1971), 313-324. Zbl0209.36401
- [16] C. Weil, On properties of derivatives, Trans. Amer. Math. Soc. 114 (1965), 363-376. Zbl0163.29604
- [17] C. Weil, On approximate and Peano derivatives, Proc. Amer. Math. Soc. 20 (1969), 487-490. Zbl0176.01103
- [18] C. Weil, A property for certain derivatives, Indiana Univ. Math. J. 23 (1973/74), 527-536. Zbl0273.26003
- [19] A. Zygmund, Trigonometric Series, 2nd ed., Cambridge Univ. Press, 1959. Zbl0085.05601

## Citations in EuDML Documents

top## NotesEmbed ?

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