 96430 Kiran M. Kolwankar, Anil D. Gangal
 Fractional differentiability of nowhere differentiable
functions and dimensions
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Sep 23, 96

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Abstract. Weierstrass's everywhere continuous but nowhere differentiable
function is shown to be locally continuously fractionally differentiable
everywhere for all orders below the `critical order' $2s$ and not so for
orders between $2s$ and $1$, where $s$, $1<s<2$, is the box dimension of
the graph of the function. This observation is consolidated in the
general result showing a direct connection between local fractional
differentiability and the box dimension/ local H\"older exponent. L\'evy
index for one dimensional L\'evy flights is shown to be the critical order
of its characteristic function. Local fractional derivatives of
multifractal signals (nonrandom functions) are shown to provide the local
H\"older exponent. It is argued that Local fractional derivatives provide
a powerful tool to analyze pointwise behavior of irregular signals.
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