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Fractional differentiability of nowhere differentiable functions and dimensions. (English) Zbl 1055.26504
Summary: 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 Hölder exponent. Lévy index for one dimensional Lévy 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 Hölder exponent. It is argued that local fractional derivatives provide a powerful tool to analyze pointwise behavior of irregular signals.

MSC:
26A33Fractional derivatives and integrals (real functions)
26A27Nondifferentiability of functions of one real variable; discontinuous derivatives