Pure point/Continuous decomposition of translation-bounded measures and diffraction

Aujogue, Jean-Baptiste

doi:10.48550/arxiv.1510.06381

Cited by 9 publications

(16 citation statements)

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“…Then, there exists a set Y ⊆ Y with the following properties: Finally, by combining Theorem 6.9 with [2] we get the following result.…”

Section: Besicovitch Almost Periodicitymentioning

confidence: 86%

“…For many examples of compatible random 1-dimensional Pisot substitutions, one gets a decomposition of the generic element ω of the hull into two measures ω 1 and ω 2 such that the diffraction of ω 1 and ω 2 , respectively, are the pure point and continuous diffraction spectrum, respectively, of ω [33,13,46]. A similar decomposition hold for 1-dimensional PV substitutions [14] and for dynamical systems of translation bounded measures [2]. It is the one of the goals of this paper to investigate this type of decomposition, at the level of measures and not autocorrelations, in more general settings.…”

Section: Introductionmentioning

confidence: 99%

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On the orthogonality of measures of different spectral type with respect to Eberlein convolution

Strungaru¹

2021

Preprint

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“…Then, there exists a set Y ⊆ Y with the following properties: Finally, by combining Theorem 6.9 with [2] we get the following result.…”

Section: Besicovitch Almost Periodicitymentioning

confidence: 86%

Section: Introductionmentioning

confidence: 99%

On the orthogonality of measures of different spectral type with respect to Eberlein convolution

Strungaru¹

2021

Preprint

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“…Let us start by recalling a result of [4]. Combining this result with Theorem 3.15 we get Theorem 6.8.…”

Section: Application To Weighted Dirac Combsmentioning

confidence: 90%

On weakly almost periodic measures

Lenz

Strungaru

2019

Trans. Amer. Math. Soc.

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We 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.

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“…We used this result to (re)derive properties of p γ, and we expect that this connection will lead to some new applications in the future. Indeed, while now we know quite a few properties of the Fourier transform of measures with Meyer set support [2,43,44,45,46,47,48,49] we know much more about fully periodic measures in LCAG (see for example [36]). Moreover, the strong admissibility of f is likely to transfer many properties from ρ to ρ f .…”

Section: Generalized Eberlein Decompositionmentioning

confidence: 99%

Fourier Transformable Measures with Meyer set support and their lift to the cut and project scheme

Strungaru¹

2021

Preprint

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In this paper, we prove that given a cut-and-project scheme pG, H, Lq and a compact window W Ď H, the natural projection gives a bijection between the Fourier transformable measures on G ˆH supported inside the strip L X pG ˆW q and the Fourier transformable measures supported inside NpWq, and relate their Fourier transforms. We use this formula to relate the Fourier transforms of the measures, and explain how one can use this relation to re-derive some known results about Fourier analysis of measures with Meyer set support.

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Pure point/Continuous decomposition of translation-bounded measures and diffraction

Cited by 9 publications

References 0 publications

On the orthogonality of measures of different spectral type with respect to Eberlein convolution

On the orthogonality of measures of different spectral type with respect to Eberlein convolution

On weakly almost periodic measures

Fourier Transformable Measures with Meyer set support and their lift to the cut and project scheme

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