A Glimpse at Mathematical Diffraction Theory

TL;DR

This paper reviews mathematical diffraction theory, focusing on the Fourier analysis of autocorrelation measures of translation bounded complex measures, highlighting key results in perfect and random systems.

Contribution

It provides a summary of rigorous mathematical results in diffraction theory, emphasizing the Fourier transform approach for analyzing diffraction measures.

Findings

Diffraction measure is the Fourier transform of autocorrelation measure.

Rigorous results are established for perfect and random systems.

The approach advances understanding of diffraction in complex measures.

Abstract

Mathematical diffraction theory is concerned with the analysis of the diffraction measure of a translation bounded complex measure $\omega$. It emerges as the Fourier transform of the autocorrelation measure of $\omega$. The mathematically rigorous approach has produced a number of interesting results in the context of perfect and random systems, some of which are summarized here.