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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Decomposition of Peano derivatives


Author: Hajrudin Fejzić
Journal: Proc. Amer. Math. Soc. 119 (1993), 599-609
MSC: Primary 26A24
DOI: https://doi.org/10.1090/S0002-9939-1993-1155596-8
MathSciNet review: 1155596
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Abstract: Let $ {\Delta '}$ be the class of all derivatives, and let $ [{\Delta '}]$ be the vector space generated by $ {\Delta '}$ and O'Malley's class $ B_1^{\ast}$. In [1] it is shown that every function in $ [{\Delta '}]$ is of the form $ {g'} + h{k'}$, where $ g,h$, and $ k$ are differentiable, and that $ f \in [{\Delta '}]$ if and only if there is a sequence of derivatives $ {v_n}$ and closed sets $ {A_n}$ such that $ \cup _{n = 1}^\infty {A_n} = {\mathbf{R}}$ and $ f = {v_n}$ on $ {A_n}$. The sequence of sets $ {A_n}$ together with the corresponding functions $ {v_n}$ is called a decomposition of $ f$. In this paper we show that every Peano derivative belongs to $ [{\Delta '}]$. Also we show that for Peano derivatives the sets $ {A_n}$ can be chosen to be perfect.


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DOI: https://doi.org/10.1090/S0002-9939-1993-1155596-8
Article copyright: © Copyright 1993 American Mathematical Society