Department of Mathematics and Statistics
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Browsing Department of Mathematics and Statistics by Subject "almost periodic measures"
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Item Modulated crystals and almost periodic measures(2020) Lee, Jeong-Yup; Lenz, Daniel; Richard, Christoph; Sing, Bernd; Strungaru, NicolaeModulated crystals and quasicrystals can simultaneously be described as modulated quasicrystals, a class of point sets introduced by de Bruijn in 1987. With appropriate modulation functions, modulated quasicrystals themselves constitute a substantial subclass of strongly almost periodic point measures. We re-analyse these structures using methods from modern mathematical diffraction theory, thereby providing a coherent view over that class. Similarly to de Bruijn's analysis, we find stability with respect to almost periodic modulations.Item On norm almost periodic measures(2019) Spindeler, Timo; Strungaru, NicolaeIn this paper, we study norm almost periodic measures on locally compact Abelian groups. First, we show that the norm almost periodicity of μ is equivalent to the equi-Bohr almost periodicity of μ∗g for all g in a fixed family of functions. Then, we show that, for absolutely continuous measures, norm almost periodicity is equivalent to the Stepanov almost periodicity of the Radon--Nikodym density.Item On norm almost periodic measures(2021) Spindeler, Timo; Strungaru, NicolaeIn this paper, we study norm almost periodic measures on locally compact Abelian groups. First, we show that the norm almost periodicity of μ is equivalent to the equi-Bohr almost periodicity of μ∗g for all g in a fixed family of functions. Then, we show that, for absolutely continuous measures, norm almost periodicity is equivalent to the Stepanov almost periodicity of the Radon–Nikodym density.Item On the Fourier analysis of measures with Meyer set support(2020) Strungaru, NicolaeIn 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.Item On the Fourier transformability of strongly almost periodic measures(2017) Strungaru, NicolaeIn this paper we characterize the Fourier transformability of a strongly almost periodic measure in terms of an integrability condition for its Fourier Bohr series. We also provide a necessary and sufficient condition for a strongly almost periodic measure to be a Fourier transform of a measure. We discuss the Fourier transformability of a measure on $\RR^d$ in terms of its Fourier transform as a tempered distribution. We conclude by looking at a large class of such measures coming from the cut and project formalism.Item On weakly almost periodic measures(2016) Lenz, Daniel; Strungaru, NicolaeWe study the diffraction and dynamical properties of translation bounded weakly almost periodic measures. We prove that the dynamical hull of a weakly almost periodic measure is a weakly almost periodic dynamical system with unique minimal component given by the hull of the strongly almost periodic component of the measure. In particular the hull is minimal if and only if the measure is strongly almost periodic and the hull is always measurably conjugate to a torus and has pure point spectrum with continuous eigenfunctions. As an application we show the stability of the class of weighted Dirac combs with Meyer set or FLC support and deduce that such measures have either trivial or large pure point respectively continuous spectrum. We complement these results by investigating the Eberlein convolution of two weakly almost periodic measures. Here, we show that it is unique and a strongly almost periodic measure. We conclude by studying the Fourier-Bohr coefficients of weakly almost periodic measures.Item On weighted Dirac combs supported inside model sets(2014) Strungaru, NicolaeIn this paper we prove that given a weakly almost periodic measure μ supported inside some model set $\Lambda (W)$ with closed window W, then the strongly almost periodic component ${{\mu }_{S}}$ and the null weakly almost periodic component μ0 are both supported inside $\Lambda (W)$. As a consequence we prove that given any translation bounded measure ω, supported inside some model set, then each of the pure point diffraction spectrum ${{\hat{\gamma }}_{{\rm pp}}}$ and the continuous diffraction spectrum ${{\hat{\gamma }}_{c}}$ is either trivial or has a relatively dense support.Item Pure point diffraction and mean, Besicovitch and Weyl almost periodicity(2020) Lenz, Daniel; Spindeler, Timo; Strungaru, NicolaeWe show that a translation bounded measure has pure point diffraction if and only if it is mean almost periodic. We then go on and show that a translation bounded measure solves what we call the phase problem if and only if it is Besicovitch almost periodic. Finally, we show that a translation bounded measure solves the phase problem independent of the underlying van Hove sequence if and only if it is Weyl almost periodic. These results solve fundamental issues in the theory of pure point diffraction and answer questions of Lagarias.