Non-periodic systems with continuous diffraction measures

TL;DR

This paper reviews and extends mathematical diffraction theory for systems with continuous spectral components, focusing on stochastic point processes and measures, and introduces new measures related to autocorrelation.

Contribution

It introduces a systematic approach using Palm measures for complex-valued random measures in diffraction theory, extending existing frameworks.

Findings

Characterization of diffraction for singular and absolutely continuous spectra

Development of a Palm measure framework for complex-valued random measures

Connection between Palm measure intensity and autocorrelation measure

Abstract

The present state of mathematical diffraction theory for systems with continuous spectral components is reviewed and extended. We begin with a discussion of various characteristic examples with singular or absolutely continuous diffraction, and then continue with a more general exposition of a systematic approach via stationary stochastic point processes. Here, the intensity measure of the Palm measure takes the role of the autocorrelation measure in the traditional approach. We furthermore introduce a `Palm-type' measure for general complex-valued random measures that are stationary and ergodic, and relate its intensity measure to the autocorrelation measure.