A Fast Fourier Transform for Fractal Approximations

TL;DR

This paper introduces a Fast Fourier Transform tailored for fractal approximations, leveraging iterated function systems to achieve efficient computation similar to classical FFT.

Contribution

It develops a novel FFT algorithm for fractal data by exploiting recursive structures from iterated function systems, reducing computational complexity.

Findings

Achieves O(N log N) complexity for fractal Fourier transforms

Uses iterated function systems to generate data points and frequencies

Recursion relations enable efficient DFT matrix computation

Abstract

We consider finite approximations of a fractal generated by an iterated function system of affine transformations on $\mathbb{R}^d$ as a discrete set of data points. Considering a signal supported on this finite approximation, we propose a Fast (Fractal) Fourier Transform by choosing appropriately a second iterated function system to generate a set of frequencies for a collection of exponential functions supported on this finite approximation. Since both the data points of the fractal approximation and the frequencies of the exponential functions are generated by iterated function systems, the matrix representing the Discrete Fourier Transform (DFT) satisfies certain recursion relations, which we describe in terms of Di\c{t}\v{a}'s construction for large Hadamard matrices. These recursion relations allow for the DFT matrix calculation to be reduced in complexity to O(N log N ), as in the case of the classical FFT.