Regularized and Fractional Taylor expansions of Holderian functions

TL;DR

This paper introduces fractional Taylor expansions for Holderian functions using fractional velocity, enabling regularization of derivatives and analysis of non-differentiable functions with singularities.

Contribution

It develops fractional-order Taylor expansions based on fractional velocity, providing new tools for analyzing and regularizing non-differentiable Holder functions.

Findings

Fractional velocity describes singular derivative behavior.

Regularized Taylor series for Holder functions are constructed.

A compound differential rule for Holder 1/2 functions is demonstrated.

Abstract

Holderian functions have strong non-linearities, which result in singularities in the derivatives. This manuscript presents several fractional-order Taylor expansions of H\"olderian functions around points of non- differentiability. These expansions are derived using the concept of a fractional velocity, which can be used to describe the singular behavior of derivatives and non-differentiable functions. Fractional velocity is defined as the limit of the difference quotient of the increment of a function and the difference of its argument raised to a fractional power. Fractional velocity can be used to regularize ordinary derivatives. To this end, it is possible to define regularized Taylor series and compound differential rules. In particular a compound differential rule for Holder 1/2 functions is demonstrated. The expansion is presented using the auxiliary concept of fractional co-variation of functions.