Local Fractional Derivatives and Fractal Functions of Several Variables

TL;DR

This paper extends the concept of local fractional derivatives to functions of multiple variables, demonstrating their usefulness in analyzing fractal properties and directional differentiability in higher dimensions.

Contribution

It introduces directional local fractional derivatives for multivariable functions and shows their application through simple examples, expanding the scope of LFD analysis.

Findings

Extended LFD to multiple variables

Demonstrated utility with simple examples

Linked LFD existence to fractal properties

Abstract

The notion of a local fractional derivative (LFD) was introduced recently for functions of a single variable. LFD was shown to be useful in studying fractional differentiability properties of fractal and multifractal functions. It was demonstrated that the local Holder exponent/ dimension was directly related to the maximum order for which LFD existed. We have extended this definition to directional-LFD for functions of many variables and demonstrated its utility with the help of simple examples.