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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On generalized Peano derivatives

Author: Cheng Ming Lee
Journal: Trans. Amer. Math. Soc. 275 (1983), 381-396
MSC: Primary 26A24; Secondary 26A39
MathSciNet review: 678358
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Abstract: A function $ F$ is said to have a generalized $ n$th Peano derivative at $ x$ if $ F$ is continuous in a neighborhood of $ x$ and if there exists a positive integer $ k$ such that a $ k$th primitive of $ F$ in the neighborhood has the $ (k + n)$th Peano derivative at $ x$; and in this case this $ (k + n)$th Peano derivative at $ x$ is proved to be independent of the integer $ k$ and the $ k$th primitives, and is called the generalized $ n$th Peano derivative of $ F$ at $ x$ which is denoted as $ {F_{[n]}}(x)$. If $ {F_{[n]}}(x)$ exists and is finite for all $ x$ in an interval, then it is shown that $ {F_{[n]}}$ shares many interesting properties that are known for the ordinary Peano derivatives. Using the generalized Peano derivatives, a notion called absolute generalized Peano derivative is studied. It is proved that on a compact interval, the absolute generalized Peano derivatives are just the generalized Peano derivatives. In particular, Laczkovich's absolute (ordinary) Peano derivatives are generalized Peano derivatives.

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Keywords: Peano derivative, generalized Peano derivative, absolute Peano derivative, absolute generalized Peano derivative, Cesaro-Perron integrals, Darboux property, Denjoy property, approximate Peano derivative
Article copyright: © Copyright 1983 American Mathematical Society