Fourier transforms on Cantor sets: A study in non-Diophantine arithmetic and calculus

TL;DR

This paper develops a Fourier transform framework on fractals with intrinsic arithmetic, enabling differentiation, integration, and complex analysis, demonstrated on a Cantor set example, applicable even to non-self-similar fractals.

Contribution

It introduces a novel Fourier transform formalism on fractals using non-Diophantine arithmetic, extending classical analysis to complex fractal structures.

Findings

Fourier transform on fractals retains basic properties

Application to a sawtooth signal on the Cantor set

Framework applicable to non-self-similar fractals

Abstract

Fractals equipped with intrinsic arithmetic lead to a natural definition of differentiation, integration and complex numbers. Applying the formalism to the problem of a Fourier transform on fractals we show that the resulting transform has all the expected basic properties. As an example we discuss a sawtooth signal on the ternary middle-third Cantor set. The formalism works also for fractals that are not self-similar.