Fractional differentiability of nowhere differentiable functions and dimensions

Area of Science:

• Fractal Geometry
• Non-differentiable Functions
• Fractional Calculus

Background:

• Weierstrass functions are classic examples of continuous but nowhere differentiable functions.
• Understanding the local behavior of such irregular functions is a significant challenge in mathematical analysis.

Purpose of the Study:

• To investigate the local fractional differentiability of Weierstrass functions.
• To establish a connection between local fractional differentiability, box dimension, and local Holder exponents.
• To explore the utility of local fractional derivatives in analyzing multifractal signals.

Main Methods:

• Analysis of local fractional differentiability orders for Weierstrass functions.
• Calculation of box dimension for the function's graph.
• Application of local fractional derivatives to multifractal signals.

Main Results:

• Weierstrass functions exhibit local continuous fractional differentiability for orders below a critical value (2-s), determined by the box dimension (s) of their graph.
• A direct relationship is demonstrated between local fractional differentiability and the box dimension/local Holder exponent.
• The Levy index of one-dimensional Levy flights corresponds to the critical order of its characteristic function.
• Local fractional derivatives accurately provide the local Holder exponent for multifractal signals.

Conclusions:

• Local fractional derivatives offer a powerful method for analyzing the pointwise behavior of irregular and multifractal signals.
• The critical order of fractional differentiability is intrinsically linked to the fractal properties (box dimension) of a function's graph.