Decomposition of Peano derivatives

Fejzić, Hajrudin

doi:10.1090/S0002-9939-1993-1155596-8

Decomposition of Peano derivatives
HTML articles powered by AMS MathViewer

by Hajrudin Fejzić

Proc. Amer. Math. Soc. 119 (1993), 599-609

DOI: https://doi.org/10.1090/S0002-9939-1993-1155596-8

PDF | Request permission

Abstract:

Let ${\Delta ’}$ be the class of all derivatives, and let $[{\Delta ’}]$ be the vector space generated by ${\Delta ’}$ and O’Malley’s class $B_1^{\ast }$. In [1] it is shown that every function in $[{\Delta ’}]$ is of the form ${g’} + h{k’}$, where $g,h$, and $k$ are differentiable, and that $f \in [{\Delta ’}]$ if and only if there is a sequence of derivatives ${v_n}$ and closed sets ${A_n}$ such that $\cup _{n = 1}^\infty {A_n} = {\mathbf {R}}$ and $f = {v_n}$ on ${A_n}$. The sequence of sets ${A_n}$ together with the corresponding functions ${v_n}$ is called a decomposition of $f$. In this paper we show that every Peano derivative belongs to $[{\Delta ’}]$. Also we show that for Peano derivatives the sets ${A_n}$ can be chosen to be perfect.

References

S. J. Agronsky, R. Biskner, A. M. Bruckner, and J. Mařík, Representations of functions by derivatives, Trans. Amer. Math. Soc. 263 (1981), no. 2, 493–500. MR 594421, DOI 10.1090/S0002-9947-1981-0594421-7
Andrew M. Bruckner, Differentiation of real functions, Lecture Notes in Mathematics, vol. 659, Springer, Berlin, 1978. MR 507448, DOI 10.1007/BFb0069821
Ernest Corominas, Contribution à la théorie de la dérivation d’ordre supérieur, Bull. Soc. Math. France 81 (1953), 177–222 (French). MR 62794, DOI 10.24033/bsmf.1442
H. William Oliver, The exact Peano derivative, Trans. Amer. Math. Soc. 76 (1954), 444–456. MR 62207, DOI 10.1090/S0002-9947-1954-0062207-1
Richard J. O’Malley, Decomposition of approximate derivatives, Proc. Amer. Math. Soc. 69 (1978), no. 2, 243–247. MR 0466446, DOI 10.1090/S0002-9939-1978-0466446-9

On generalized derivatives

3

J. Mařík, Derivatives and closed sets, Acta Math. Hungar. 43 (1984), no. 1-2, 25–29. MR 731958, DOI 10.1007/BF01951320
S. Verblunsky, On the Peano derivatives, Proc. London Math. Soc. (3) 22 (1971), 313–324. MR 285678, DOI 10.1112/plms/s3-22.2.313
Clifford E. Weil, On properties of derivatives, Trans. Amer. Math. Soc. 114 (1965), 363–376. MR 176007, DOI 10.1090/S0002-9947-1965-0176007-2
Clifford E. Weil, On approximate and Peano derivatives, Proc. Amer. Math. Soc. 20 (1969), 487–490. MR 233944, DOI 10.1090/S0002-9939-1969-0233944-7
Clifford E. Weil, A property for certain derivatives, Indiana Univ. Math. J. 23 (1973/74), 527–536. MR 335703, DOI 10.1512/iumj.1973.23.23044
A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776