Fractional differentiability of nowhere differentiable functions and dimensions

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 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.