Fractional variation of H\"olderian functions

TL;DR

This paper explores local fractional variation operators and their application to H"olderian functions, revealing singular behaviors and connections to derivatives, with examples including fractional variation of sequences and delta functions.

Contribution

It introduces and analyzes the properties of fractal variation operators for H"olderian functions, linking their limits to derivatives and illustrating with diverse examples.

Findings

Fractal variation operators characterize H"olderian functions locally.

Certain functions exhibit singular behavior under fractal variation in the infinitesimal limit.

Fractional variation of sequences can lead to the Dirac delta-function.

Abstract

The paper demonstrates the basic properties of the local fractional variation operators (termed fractal variation operators). The action of the operators is demonstrated for local characterization of Holderian functions. In particular, it is established that a class of such functions exhibits singular behavior under the action of fractal variation operators in infinitesimal limit. The link between the limit of the fractal variation of a function and its derivative is demonstrated. The paper presents a number of examples, including the calculation of the fractional variation of Cauchy sequences leading to the Dirac's delta-function.