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

2021/05/07 by Timo Spindeler, Nicolae Strungaru, Spindeler, Timo +1
Mathematics · #43A05 #43A25 #52C23 #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Approximation Theory and Sequence Spaces #FOS: Mathematics #Functional Analysis (math.FA)

paper · pdf · doi:10.48550/arxiv.2105.03382

openalex publication_date 2021/05/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 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.

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