96-430 Kiran M. Kolwankar, Anil D. Gangal
Fractional differentiability of nowhere differentiable functions and dimensions (23K, gzipped uuencoded LaTeX) 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' $2-s$ and not so for orders between $2-s$ 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 (non-random 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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