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.