Computing with almost periodic functions

TL;DR

This paper introduces a practical, group-theoretic method for discrete Fourier analysis of functions on quasicrystals, enabling uniform approximation over infinite spaces using finite group structures.

Contribution

It develops a novel computational approach for Fourier analysis on almost periodic sets, leveraging the theory of quasicrystals and diffraction within a group-theoretic framework.

Findings

Method effectively approximates functions on quasicrystals

Applicable to all quasicrystals modeled on cut and project formalism

Demonstrated with Fibonacci quasicrystal examples

Abstract

The paper develops a method for discrete computational Fourier analysis of functions defined on quasicrystals and other almost periodic sets. A key point is to build the analysis around the emerging theory of quasicrystals and diffraction in the setting on local hulls and dynamical systems. Numerically computed approximations arising in this way are built out of the Fourier module of the quasicrystal in question, and approximate their target functions uniformly on the entire infinite space. The methods are entirely group theoretical, being based on finite groups and their duals, and they are practical and computable. Examples of functions based on the standard Fibonacci quasicrystal serve to illustrate the method (which is applicable to all quasicrystals modeled on the cut and project formalism).