Regularized and Fractional Taylor expansions of Holderian functions

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.