Computing with almost periodic functions

Area of Science:

• Mathematical Physics
• Harmonic Analysis
• Materials Science

Background:

• Quasicrystals and almost periodic sets exhibit complex structures challenging traditional Fourier analysis.
• Existing methods often struggle with the non-periodic nature of these materials.
• The emerging theory of quasicrystals and diffraction provides new frameworks for analysis.

Purpose of the Study:

• To develop a discrete computational Fourier analysis method for functions on quasicrystals.
• To leverage local hulls and dynamical systems within quasicrystal theory for diffraction analysis.
• To create practical, computable approximations for functions on infinite quasicrystal structures.

Main Methods:

• Utilizing group theory and finite group duals for a group-theoretical approach.
• Building analysis around the Fourier module of the specific quasicrystal.
• Employing local hulls and dynamical systems in the analysis framework.

Main Results:

• Development of a novel discrete computational Fourier analysis method.
• Achieving uniform approximations of target functions across the entire infinite space.
• Demonstration of practical and computable group-theoretical methods.

Conclusions:

• The developed method offers a robust way to perform Fourier analysis on quasicrystals.
• The approach is applicable to all quasicrystals modeled using the cut-and-project formalism.
• The group-theoretical foundation ensures practicality and computability for real-world applications.