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Some Comments on the Inverse Problem of Pure Point Diffraction

  • Venta Terauds
  • Michael Baake
Conference paper

Abstract

In a recent paper arXiv:1111.3617, Lenz and Moody presented a method for constructing families of real solutions to the inverse problem for a given pure point diffraction measure. Applying their technique and discussing some possible extensions, we present, in a non-technical manner, some examples of homometric structures.

Keywords

Point Process Diffraction Measure Compact Abelian Group Pure Point Lower Path 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

This work was supported by the German Research Council (DFG), via the CRC 701, and by the RCM2, at the University of Bielefeld.

References

  1. 1.
    Baake M, Grimm U (2007) Homometric model sets and window covariograms. Z Kristallogr 222:54–58 CrossRefGoogle Scholar
  2. 2.
    Baake M, Grimm U (2012) Mathematical diffraction of aperiodic structures. Chem Soc Rev 41:6821–6843 CrossRefGoogle Scholar
  3. 3.
    Baake M, Grimm U (2013) Theory of aperiodic order: a mathematical invitation. Cambridge University Press, Cambridge, to appear Google Scholar
  4. 4.
    Baake M, Moody RV (2004) Weighted Dirac combs with pure point diffraction. J Reine Angew Math 573:61–94 Google Scholar
  5. 5.
    Córdoba A (1989) Dirac combs. Lett Math Phys 17:191–196 CrossRefGoogle Scholar
  6. 6.
    Gähler F, Klitzing R (1995) The diffraction pattern of self-similar tilings. In: Moody RV (ed) The mathematics of long- range aperiodic order. NATO ASI series C, vol 489. Kluwer, Dordrecht, pp 141–174 Google Scholar
  7. 7.
    Grimm U, Baake M (2008) Homometric point sets and inverse problems. Z Kristallogr 223:777–781 CrossRefGoogle Scholar
  8. 8.
    Grünbaum FA, Moore CC (1995) The use of higher-order invariants in the determination of generalized Patterson cyclotomic sets. Acta Crystallogr, Ser A 51:310–323 CrossRefGoogle Scholar
  9. 9.
    Hof A (1995) On diffraction by aperiodic structures. Commun Math Phys 169:25–43 CrossRefGoogle Scholar
  10. 10.
    Hof A (1995) Diffraction by aperiodic structures. In: Moody RV (ed) The mathematics of long-range aperiodic order. NATO ASI series C, vol 489. Kluwer, Dordrecht, pp 239–268 Google Scholar
  11. 11.
    Lenz D, Moody RV Stationary processes with pure point diffraction. Preprint. arXiv:1111.3617
  12. 12.
    Terauds V (2013) The inverse problem of pure point diffraction—examples and open questions, in preparation Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität BielefeldBielefeldGermany

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