On absolute Peano derivatives
type: Item
instance of: scholarly article
title: On absolute Peano derivatives
zbMATH Open document ID: 0535.26003
published in: Real Analysis Exchange
publication date: 1983-01-01T00:00:00Z
review text: In a recent survey [Real Anal. Exch. 7, 5-23 (1982; preceding review)], \textit{M. J. Evans} and \textit{C. E. Weil} asked if the absolute Peano derivative of \textit{M. Laczkovich} [Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 21, 83-97 (1978; Zbl 0425.26005)] has the -M,M and Z properties, properties enjoyed by most other derivatives. The present author has defined generalized Peano derivatives in Trans. Am. Math. Soc. 275, 381-396 (1983; Zbl 0506.26006). A function f is said to have generalized n-th Peano derivative at t if f is continuous in some neighbourhood of t and if for some positive integer k the k-th primitive of f has a \((k+n)\)-th Peano derivative at t. In that paper the author has shown also that every absolute Peano derivative is a generalized Peano derivative. In the present article the author proves that generalized Peano derivatives have both of the above properties, thus answers the question of Evans and Weil positively. His proof uses basic results from his article quoted above, certain ideas of Mařík, O'Malley and Weil, and the extension to generalized Peano derivatives of the work of Sargent on Peano derivatives.
Mathematics Subject Classification ID: 26A24
zbMATH DE Number: 3849527
zbMATH Keywords: absolute Peano derivative
zbMATH Keywords: -M,M property
zbMATH Keywords: Z property
author: Cheng Ming Lee
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3849527