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Aperiodic order and spherical diffraction, III: The shadow transform and the diffraction formula

Björklund, Michael; Hartnick, Tobias; Pogorzelski, Felix

Abstract:
We define spherical diffraction measures for a wide class of weighted point sets in commutative spaces, i.e. proper homogeneous spaces associated with Gelfand pairs. In the case of the hyperbolic plane we can interpret the spherical diffraction measure as the Mellin transform of the auto-correlation distribution. We show that uniform regular model sets in commutative spaces have a pure point spherical diffraction measure. The atoms of this measure are located at the spherical automorphic spectrum of the underlying lattice, and the diffraction coefficients can be characterized abstractly in terms of the so-called shadow transform of the characteristic functions of the window. In the case of the Heisenberg group we can give explicit formulas for these diffraction coefficients in terms of Bessel and Laguerre functions.


Verlagsausgabe §
DOI: 10.5445/IR/1000139159
Veröffentlicht am 19.10.2021
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Algebra und Geometrie (IAG)
Publikationstyp Zeitschriftenaufsatz
Publikationsjahr 2021
Sprache Englisch
Identifikator ISSN: 0022-1236, 1096-0783
KITopen-ID: 1000139159
Erschienen in Journal of Functional Analysis
Verlag Elsevier
Band 281
Heft 12
Seiten Art.-Nr.: 109265
Nachgewiesen in Scopus
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