On properties of the approximate Peano derivatives

Babcock, Bruce S.

doi:10.1090/S0002-9947-1975-0414803-0

On properties of the approximate Peano derivatives
HTML articles powered by AMS MathViewer

by Bruce S. Babcock PDF

Trans. Amer. Math. Soc. 212 (1975), 279-294 Request permission

Abstract:

The notion of kth approximate Peano differentiation not only generalizes kth ordinary differentiation but also kth Peano differentiation and kth ${L_p}$ differentiation. Recently, M. Evans has shown that a kth approximate Peano derivative at least shares with these other derivatives the property of belonging to Baire class one. In this paper the author extends the properties possessed by a kth approximate Peano derivative by showing that it is like the above derivatives in that it also possesses the following properties: Darboux, Denjoy, Zahorski, and a new property stronger than the Zahorski property, Property Z.

References

Sur une propriété des fonctions dérivées exactes

18

Michael J. Evans, $L\,_{p}$ derivatives and approximate Peano derivatives, Trans. Amer. Math. Soc. 165 (1972), 381–388. MR 293030, DOI 10.1090/S0002-9947-1972-0293030-1
Casper Goffman and C. J. Neugebauer, On approximate derivatives, Proc. Amer. Math. Soc. 11 (1960), 962–966. MR 118792, DOI 10.1090/S0002-9939-1960-0118792-2
Solomon Marcus, On a theorem of Denjoy and on approximate derivative, Monatsh. Math. 66 (1962), 435–440. MR 151558, DOI 10.1007/BF01298240
C. J. Neugebauer, Darboux functions of Baire class one and derivatives, Proc. Amer. Math. Soc. 13 (1962), 838–843. MR 143850, DOI 10.1090/S0002-9939-1962-0143850-8
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

Sur la dérivée approximative exacte

4

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, A property for certain derivatives, Indiana Univ. Math. J. 23 (1973/74), 527–536. MR 335703, DOI 10.1512/iumj.1973.23.23044
Z. Zahorski, Sur la première dérivée, Trans. Amer. Math. Soc. 69 (1950), 1–54 (French). MR 37338, DOI 10.1090/S0002-9947-1950-0037338-9
A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776