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In this paper, we study the class of tempered distributions whose Fouriertransform is a translation bounded measure and show that each such distributionin $\mathbb{R}^d$ has order at most $2d$. We show the existence of thegeneralized Eberlein decomposition within this class of distributions, and itscompatibility with all previous Eberlein decompositions. The generalizedEberlein decomposition for Fourier transformable measures and properties of itscomponents are discussed. Lastly, we take a closer look at the absolutelycontinuous spectrum of measures supported on Meyer sets.

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