A Commutative Alternative to Fractional Calculus on Continuous Functions

TL;DR

This paper introduces a new commutative operator extending fractional calculus concepts from real analytic functions to all real continuous functions, broadening the scope of fractional calculus techniques.

Contribution

It develops a commutative operator for continuous functions, expanding fractional calculus beyond real analytic functions and maintaining key algebraic properties.

Findings

Operator commutes with itself on continuous functions

Extension from real analytic to continuous functions achieved

Potential applications in analysis and differential equations

Abstract

In [1], an operator was introduced which acts parallel to the Riemann-Liouville differintegral on a transformation of the space of real analytic functions and commutes with itself. This paper aims to extend the technique - and its defining characteristic, commutativity - to all real continuous functions, up to the degree to which they are differentiable.