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Transactions of the American Mathematical Society

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



A characterization of the Peano derivative

Author: J. Marshall Ash
Journal: Trans. Amer. Math. Soc. 149 (1970), 489-501
MSC: Primary 26.43
MathSciNet review: 0259041
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Abstract: For each choice of parameters $ \{ {a_i},{b_i}\} ,i = 0,1, \ldots ,n + e$, satisfying certain simple conditions, the expression

$\displaystyle \mathop {\lim }\limits_{h \to 0} {h^{ - n}}\sum\limits_{i = 0}^{n + e} {{a_i}f(x + {b_i}h)} $

yields a generalized nth derivative. A function f has an nth Peano derivative at x if and only if all the members of a certain subfamily of these nth derivatives exist at x. The result holds for the corresponding $ {L^p}$ derivatives. A uniformity lemma in the proof (Lemma 2) may be of independent interest.

Also, a new generalized second derivative is introduced which differentiates more functions than the ordinary second derivative but fewer than the second Peano derivative.

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Keywords: Peano derivative, generalized derivative, Riemann derivative, function of one real variable, $ {L^p}$ derivative
Article copyright: © Copyright 1970 American Mathematical Society