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Stationary processes and pure point diffraction

Published online by Cambridge University Press:  04 July 2016

DANIEL LENZ  and
ROBERT V. MOODY
DANIEL LENZ
Affiliation:
Mathematisches Institut, Fakultät für Mathematik und Informatik, Friedrich-Schiller-Universität Jena, Jena D-07743, Germany email daniel.lenz@uni-jena.de
ROBERT V. MOODY
Affiliation:
Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3P4, Canada email rmoody@uvic.ca
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Abstract

We consider the construction and classification of some new mathematical objects, called ergodic spatial stationary processes, on locally compact abelian groups. These objects provide a natural and very general setting for studying diffraction and the famous inverse problems associated with it. In particular, we can construct complete families of solutions to the inverse problem from any given positive pure point measure that is chosen to be the diffraction. In this case these processes can be classified by the dual of the group of relators based on the set of Bragg peaks, and this gives an abstract solution to the homometry problem for pure point diffraction.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 8 , December 2017 , pp. 2597 - 2642
Copyright
© Cambridge University Press, 2016 

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