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Browsing by Author "Spindeler, Timo"

Now showing 1 - 9 of 9
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    Diffraction of compatible random substitutions in one dimension
    (2018) Baake, Michael; Spindeler, Timo; Strungaru, Nicolae
    As 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.
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    A note on measures vanishing at infinity
    (2019) Strungaru, Nicolae; Spindeler, Timo
    In 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.
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    On norm almost periodic measures
    (2019) Spindeler, Timo; Strungaru, Nicolae
    In 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.
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    On norm almost periodic measures
    (2021) Spindeler, Timo; Strungaru, Nicolae
    In 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.
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    On the (dis)continuity of the Fourier transform of measures
    (2021) Spindeler, Timo; Strungaru, Nicolae
    In 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.
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    On the (dis)continuity of the Fourier transform of measures
    (2020) Spindeler, Timo; Strungaru, Nicolae
    In 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.
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    Pure point diffraction and mean, Besicovitch and Weyl almost periodicity
    (2020) Lenz, Daniel; Spindeler, Timo; Strungaru, Nicolae
    We 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.
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    Pure point spectrum for dynamical systems and mean almost periodicity
    (2020) Lenz, Daniel; Spindeler, Timo; Strungaru, Nicolae
    We consider metrizable ergodic topological dynamical systems over locally compact, σ-compact abelian groups. We study pure point spectrum via suitable notions of almost periodicity for the points of the dynamical system. More specifically, we characterize pure point spectrum via mean almost periodicity of generic points. We then go on and show how Besicovitch almost periodic points determine both eigenfunctions and the measure in this case. After this, we characterize those systems arising from Weyl almost periodic points and use this to characterize weak and Bohr almost periodic systems. Finally, we consider applications to aperiodic order.
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    Pure point spectrum for dynamical systems and mean, Besicovitch and Weyl almost periodicity
    (2023) Lenz, Daniel; Spindeler, Timo; Strungaru, Nicolae
    We consider metrizable ergodic topological dynamical systems over locally compact, σ - compact abelian groups. We study pure point spectrum via suitable notions of almost periodicity for the points of the dynamical system. More specifically, we characterize pure point spectrum via mean almost periodicity of generic points. We then go on and show how Besicovitch almost periodic points determine both eigenfunctions and the measure in this case. After this, we characterize those systems arising from Weyl almost periodic points and use this to characterize weak and Bohr almost periodic systems. Finally, we consider applications to aperiodic order.