Simple fractal calculus from fractal arithmetic

TL;DR

This paper introduces a simplified non-Newtonian calculus based on fractal arithmetic, enabling easier derivatives and integrals on fractals like the Sierpiński set, with practical applications such as Fourier transforms.

Contribution

It presents a new, simpler framework for calculus on fractals using non-Diophantine arithmetic, applicable where traditional methods are complex or fail.

Findings

Simpler definitions of derivatives and integrals on fractals

Successful Fourier transform on Sierpiński set domain

Applicable to all continuum-cardinality fractals

Abstract

Non-Newtonian calculus that starts with elementary non-Diophantine arithmetic operations of a Burgin type is applicable to all fractals whose cardinality is continuum. The resulting definitions of derivatives and integrals are simpler from what one finds in the more traditional literature of the subject, and they often work in the cases where the standard methods fail. As an illustration, we perform a Fourier transform of a real-valued function with Sierpi\'nski-set domain. The resulting formalism is as simple as the usual undergraduate calculus.