Measures with locally finite support and spectrum

Area of Science:

• Mathematical Physics
• Harmonic Analysis
• Measure Theory

Background:

• Exploration of mathematical measures with specific properties is crucial in fields like signal processing and crystallography.
• Existing frameworks often struggle to reconcile properties like discrete Fourier transforms with non-periodic structures.
• Generalized Dirac combs are standard models for periodic structures, but understanding aperiodic order requires different approaches.

Purpose of the Study:

• To construct mathematical measures (μ) on R(n) that possess three key properties simultaneously.
• To demonstrate that these measures are sums of weighted Dirac masses, with Fourier transforms also being sums of weighted Dirac masses.
• To ensure these measures are not generalized Dirac combs, thus providing examples of aperiodic order.

Main Methods:

• Construction of novel mathematical measures on R(n).
• Analysis of the properties of these measures, specifically their structure as sums of weighted Dirac masses.
• Investigation of the Fourier transform of these measures to confirm it also exhibits the desired structure.

Main Results:

• Successfully constructed measures (μ) satisfying all three specified properties.
• The constructed measures are sums of weighted Dirac masses, and their Fourier transforms are also sums of weighted Dirac masses.
• These measures are explicitly not generalized Dirac combs, offering new examples of aperiodic structures.

Conclusions:

• The study provides simple yet surprising examples of measures with compatible, seemingly conflicting properties.
• These findings introduce unusual Poisson's summation formulas and differ significantly from quasicrystal structures.
• The results offer a novel perspective on understanding and modeling aperiodic order in mathematical and physical systems.