Stationary processes and pure point diffraction. (English) Zbl 1380.37020
Summary: 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.
MSC:
| 37A60 | Dynamical aspects of statistical mechanics |
| 37A50 | Dynamical systems and their relations with probability theory and stochastic processes |
| 78A45 | Diffraction, scattering |
| 78A46 | Inverse problems (including inverse scattering) in optics and electromagnetic theory |
| 81U40 | Inverse scattering problems in quantum theory |
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