On the Fourier Analysis of Measures with Meyer Set Support

Abstract

In this paper we show the existence of the generalized Eberlein decomposition for Fourier transformable measures with Meyer set support. We prove that each of the three components is also Fourier transformable and has Meyer set support. We obtain that each of the pure point, absolutely continuous and singular continuous components of the Fourier transform is a strong almost periodic measure, and hence is either trivial or has relatively dense support. We next prove that the Fourier transform of a measure with Meyer set support is norm almost periodic, and hence so is each of the pure point, absolutely continuous and singular continuous components. We show that a measure with Meyer set support is Fourier transformable if and only if it is a linear combination of positive definite measures, which can be chosen with Meyer set support, solving a particular case of an open problem. We complete the paper by discussing some applications to the diffraction of weighted Dirac combs with Meyer set support.

arXiv comment29 pages

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Cited by in corpus (14)

• Pure Point Diffraction and Mean, Besicovitch and Weyl Almost Periodicity
• On norm almost periodic measures
• Modulated crystals and almost periodic measures
• On arithmetic progressions in model sets
• Pure point measures with sparse support and sparse Fourier--Bohr support
• Inflation versus projection sets in aperiodic systems: The role of the window in averaging and diffraction
• Model sets with precompact Borel windows
• Stepanov and Weyl almost periodic functions on locally compact Abelian groups
• Eberlein decomposition for PV inflation systems
• On eigenmeasures under Fourier transform
• Why do Meyer sets diffract?
• Semi-measures and their Fourier transform
• Fourier Transformable Measures with Meyer set support and their lift to the cut and project scheme
• Diffraction of the primes and other sets of zero density