Open AccessComputing with almost periodic functions
Author(s) -
Moody R. V.,
Nesterenko M.,
Patera J.
Publication year - 2008
Publication title -
acta crystallographica section a
Language(s) - English
Resource type - Journals
eISSN - 1600-5724
pISSN - 0108-7673
DOI - 10.1107/s0108767308025440
Subject(s) - quasicrystal , dual polyhedron , fibonacci number , fourier transform , mathematics , formalism (music) , fourier series , fourier analysis , mathematical analysis , pure mathematics , discrete mathematics , geometry , art , musical , visual arts
This 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).