### Browsing by Author "Strungaru, Nicolae"

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Item Almost periodic measures and Bragg diffraction(2013) Strungaru, NicolaeIn this paper we prove that the cone $\mathcal {PDS}(G)$ of positive, positive definite, discrete and strong almost periodic measures over a σ-compact, locally compact Abelian group G has an interesting property: given any positive and positive definite measure μ smaller than some measure in $\mathcal {PDS}(G)$, the strong almost periodic part μS of μ is also in $\mathcal {PDS}(G)$. We then use this result to prove that given a positive-weighted Dirac comb ω with finite local complexity and pure point diffraction, any positive Dirac comb less than ω has either a trivial Bragg spectrum or a relatively dense set of Bragg peaks.Item Almost periodic measures and long-range order in Meyer sets(2005) Strungaru, NicolaeThe main result of this paper is that the diffraction pattern of any Meyer set with a well-defined autocorrelation has a relatively dense set of Bragg peaks. In the second part of the paper we provide a necessary and sufficient condition for a positive pure point measure to have a continuous Fourier transform. In particular, one can get a necessary and sufficient condition for a point set to have no Bragg peaks in its diffraction.Item Circular symmetry of pinwheel diffraction(2006) Strungaru, Nicolae; Moody, R. V.; Postnikoff, D.A method is given for explicitly determining the autocorrelation of the pinwheel tiling by use of the substitution system generating the tiling. Using this a new proof of the circular symmetry of the diffraction of the pinwheel tiling is given.Item Diffraction of compatible random substitutions in one dimension(2018) Baake, Michael; Spindeler, Timo; Strungaru, NicolaeAs a guiding example, the diffraction measure of a random local mixture of the two classic Fibonacci substitutions is determined and reanalysed via self-similar measures of Hutchinson type, defined by a finite family of contractions. Our revised approach yields explicit formulas for the pure point and the absolutely continuous parts, as well as a proof for the absence of singular continuous components. This approach is then extended to the family of random noble means substitutions and, as an example with an underlying 2-adic structure, to a locally randomised version of the period doubling chain. As a first step towards a more general approach, we interpret our findings in terms of a disintegration over the Kronecker factor, which is the maximal equicontinuous factor of a covering model set.Item Diffraction of fully Euclidean model sets(2021) Klick, Anna; Strungaru, NicolaeWe provide a elementary proof, using only the Poisson summation formula and theory of tempered distributions, of the well-known fact that fully Euclidean regular model sets produce a pure point diffraction measure.Item Diffraction theory and almost periodic distributions(2016) Strungaru, Nicolae; Terauds, VentaWe introduce and study the notions of translation bounded tempered distributions, and autocorrelation for a tempered distrubution. We further introduce the spaces of weakly, strongly and null weakly almost periodic tempered distributions and show that for weakly almost periodic tempered distributions the Eberlein decomposition holds. For translation bounded measures all these notions coincide with the classical ones. We show that tempered distributions with measure Fourier transform are weakly almost periodic and that for this class, the Eberlein decomposition is exactly the Fourier dual of the Lesbegue decomposition, with the Fourier-Bohr coefficients specifying the pure point part of the Fourier transform. We complete the project by looking at few interesting examples.Item Eberlein decomposition for PV inflation systems(2020) Baake, Michael; Strungaru, NicolaeThe Dirac combs of primitive Pisot--Vijayaraghavan (PV) inflations on the real line or, more generally, in Rd are analysed. We construct a mean-orthogonal splitting for such Dirac combs that leads to the classic Eberlein decomposition on the level of the pair correlation measures, and thus to the separation of pure point versus continuous spectral components in the corresponding diffraction measures. This is illustrated with two guiding examples, and an extension to more general systems with randomness is outlined.Item Eberlein decomposition for PV inflation systems(2021) Baake, Michael; Strungaru, NicolaeThe Dirac combs of primitive Pisot–Vijayaraghavan (PV) inflations on the real line or, more generally, in Rd are analysed. We construct a mean-orthogonal splitting for such Dirac combs that leads to the classic Eberlein decomposition on the level of the pair correlation measures, and thus to the separation of pure point versus continuous spectral components in the corresponding diffraction measures. This is illustrated with two guiding examples, and an extension to more general systems with randomness is outlined.Item Fourier transformable measures with weak Meyer set support and their lift to the cut-and-project scheme(2023) Strungaru, NicolaeIn this paper, we prove that given a cut-and-project scheme (G,H,L) and a compact window W⊆H, the natural projection gives a bijection between the Fourier transformable measures on G×H supported inside the strip L∩(G×W) and the Fourier transformable measures on G supported inside ⋏(W). We provide a closed formula relating the Fourier transform of the original measure and the Fourier transform of the projection. We show that this formula can be used to re-derive some known results about Fourier analysis of measures with weak Meyer set support.Item Leptin densities in amenable groups(2022) Pogorzelski, Felix; Richard, Christoph; Strungaru, NicolaeConsider a positive Borel measure on a locally compact group. We define a notion of uniform density for such a measure, which is based on a group invariant introduced by Leptin in 1966. We then restrict to unimodular amenable groups and to translation bounded measures. In that case our density notion coincides with the well-known Beurling density from Fourier analysis, also known as Banach density from dynamical systems theory. We use Leptin densities for a geometric proof of the model set density formula, which expresses the density of a uniform regular model set in terms of the volume of its window, and for a proof of uniform mean almost periodicity of such model sets.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 A note on measures vanishing at infinity(2019) Strungaru, Nicolae; Spindeler, TimoIn this paper, we review the basic properties of measures vanishing at infinity and prove a version of the Riemann–Lebesgue lemma for Fourier transformable measures.Item A note on tempered measures(2023) Baake, Michael; Strungaru, NicolaeThe relation between tempered distributions and measures is analysed and clarified. While this is straightforward for positive measures, it is surprisingly subtle for signed or complex measures.Item Note on the set of Bragg peaks with high intensity(2016) Strungaru, Nicolae; Lenz, DanielWe consider diffraction of Delone sets in Euclidean space. We show that the set of Bragg peaks with high intensity is always Meyer (if it is relatively dense). We use this to provide a new characterization for Meyer sets in terms of positive and positive definite measures. Our results are based on a careful study of positive definite measures, which may be of interest in its own right.Item On arithmetic progressions in model sets(2020) Klick, Anna; Strungaru, Nicolae; Tcaciuc, Adi; Strungaru, NicolaeIn this project we show the existence of arbitrary length arithmetic progressions in model sets and Meyer sets in the Euclidean d-space. We prove a van der Waerden type theorem for Meyer sets. We show that pure point subsets of Meyer sets with positive density and pure point diffraction contain arithmetic progressions of arbitrary length.Item On higher dimensional arithmetic progressions in Meyer sets(2023) Klick, Anna; Strungaru, NicolaeIn this paper we study the existence of higher dimensional arithmetic progression in Meyer sets. We show that the case when the ratios are linearly dependent over $\ZZ$ is trivial, and focus on arithmetic progressions for which the ratios are linearly independent. Given a Meyer set Λ and a fully Euclidean model set $\oplam(W)$ with the property that finitely many translates of $\oplam(W)$ cover Λ, we prove that we can find higher dimensional arithmetic progressions of arbitrary length with k linearly independent ratios in Λ if and only if k is at most the rank of the $\ZZ$-module generated by $\oplam(W)$. We use this result to characterize the Meyer sets which are subsets of fully Euclidean model sets.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 (dis)continuity of the Fourier transform of measures(2021) Spindeler, Timo; Strungaru, NicolaeIn this paper, we will study the continuity of the Fourier transform of measures with respect to the vague topology. We show that the Fourier transform is vaguely discontinuous on R, but becomes continuous when restricting to a class of Fourier transformable measures such that either the measures, or their Fourier transforms are equi-translation bounded. We discuss continuity of the Fourier transform in the product and norm topology. We show that vague convergence of positive definite measures implies the equi translation boundedness of the Fourier transforms, which explains the continuity of the Fourier transform on the cone of positive definite measures. In the appendix, we characterize vague precompactness of a set of measures in arbitrary LCAG, and the necessity of second countability property of a group for defining the autocorrelation measure.Item On the (dis)continuity of the Fourier transform of measures(2020) Spindeler, Timo; Strungaru, NicolaeIn this paper, we will study the continuity of the Fourier transform of measures with respect to the vague topology. We show that the Fourier transform is vaguely discontinuous on R, but becomes continuous when restricting to a class of Fourier transformable measures such that either the measures, or their Fourier transforms are equi-translation bounded. We discuss continuity of the Fourier transform in the product and norm topology. We show that vague convergence of positive definite measures implies the equi translation boundedness of the Fourier transforms, which explains the continuity of the Fourier transform on the cone of positive definite measures. In the appendix, we characterize vague precompactness of a set a measures in arbitrary LCAG, and the necessity of second countability property of a group for defining the autocorrelation measure.