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Tempered distributions with translation bounded measure as Fourier transform and the generalized Eberlein decomposition

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  • Timo Spindeler
  • Nicolae Strungaru

Abstract

In this paper, we study the class of tempered distributions whose Fourier transform is a translation bounded measure and show that each such distribution in Rd${\mathbb {R}}^d$ has order at most 2d. We show the existence of the generalized Eberlein decomposition within this class of distributions, and its compatibility with all previous Eberlein decompositions. The generalized Eberlein decomposition for Fourier transformable measures and properties of its components are discussed. Lastly, we take a closer look at the absolutely continuous spectrum of measures supported on Meyer sets.

Suggested Citation

  • Timo Spindeler & Nicolae Strungaru, 2024. "Tempered distributions with translation bounded measure as Fourier transform and the generalized Eberlein decomposition," Mathematische Nachrichten, Wiley Blackwell, vol. 297(2), pages 716-740, February.
    • Handle: RePEc:bla:mathna:v:297:y:2024:i:2:p:716-740
    DOI: 10.1002/mana.202100658
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