On generalized Peano derivatives

Lee, Cheng Ming

doi:10.1090/S0002-9947-1983-0678358-2

On generalized Peano derivatives
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Trans. Amer. Math. Soc. 275 (1983), 381-396 Request permission

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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