### Browsing by Author "Strungaru, Nicolae"

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- ItemA 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.
- ItemA short guide to pure point diffraction in cut-and-project sets(2017) Richard, Christoph; Strungaru, NicolaeWe briefly review the diffraction of quasicrystals and then give an elementary alternative proof of the diffraction formula for regular cut-and-project sets, which is based on Bochner's theorem from Fourier analysis. This clarifies a common view that the diffraction of a quasicrystal is determined by the diffraction of its underlying lattice. To illustrate our approach, we will also treat a number of well-known explicitly solvable examples.
- ItemAlmost 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.
- ItemAlmost 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.
- ItemCircular 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.
- ItemDiffraction 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.
- ItemDiffraction 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.
- ItemDiffraction 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.
- ItemEberlein 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.
- ItemEberlein 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.
- ItemFourier 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.
- ItemModulated 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.
- ItemNote 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.
- ItemOn 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.
- ItemOn 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.
- ItemOn 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.
- ItemOn 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.
- ItemOn 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.
- ItemOn the Bragg diffraction spectra of a Meyer set(2013) Strungaru, NicolaeMeyer sets have a relatively dense set of Bragg peaks, and for this reason they may be considered as basic mathematical examples of (aperiodic) crystals. In this paper we investigate the pure point part of the diffraction of Meyer sets in more detail. The results are of two kinds. First, we show that, given a Meyer set and any positive intensity a less than the maximum intensity of its Bragg peaks, the set of Bragg peaks whose intensity exceeds a is itself a Meyer set (in the Fourier space). Second, we show that if a Meyer set is modified by addition and removal of points in such a way that its density is not altered too much (the allowable amount being given explicitly as a proportion of the original density), then the newly obtained set still has a relatively dense set of Bragg peaks.
- ItemOn 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.