Decomposition of Peano derivatives

Fejzić, Hajrudin

doi:10.1090/S0002-9939-1993-1155596-8

Decomposition of Peano derivatives
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Proc. Amer. Math. Soc. 119 (1993), 599-609 Request permission

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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On generalized derivatives

3

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