Almost periodic measures

Lamadrid, Jesús Gil de; Argabright, L. N.

doi:10.1090/memo/0428

Cited by 68 publications

(108 citation statements)

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“…The concept of almost periodicity of measures in the theory of quasicrystals has appeared before [37]. In fact, during the work for this article, we became aware that the Bohr theory has already been completely generalized to measures in a fundamental but rather neglected paper of Gil de Lamadrid and Argabright [19]. In some sense, this work already partly contains the main result of this paper as will become clear in Section 9.…”

Section: Introductionmentioning

confidence: 85%

“…Following [19], we will denote the space of translation bounded complex measures on G by M ∞ (G). For most of this paper, we consider M ∞ (G) equipped with the vague topology.…”

Section: General Setup and Class Of Measuresmentioning

confidence: 99%

“…Such a characterization can be given on the basis of [19], however for the price of losing the constructive part of our above analysis.…”

Section: Extensions: Almost Periodic Measuresmentioning

confidence: 99%

See 2 more Smart Citations

Weighted Dirac combs with pure point diffraction

Baake¹,

Moody²

2004

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

140

374

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A class of translation bounded complex measures, which have the form of weighted Dirac combs, on locally compact Abelian groups is investigated. Given such a Dirac comb, we are interested in its diffraction spectrum which emerges as the Fourier transform of the autocorrelation measure. We present a sufficient set of conditions to ensure that the diffraction measure is a pure point measure. Simultaneously, we establish a natural link to the cut and project formalism and to the theory of almost periodic measures. Our conditions are general enough to cover the known theory of model sets, but also to include examples such as the visible lattice points.

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Section: Introductionmentioning

confidence: 85%

“…Following [19], we will denote the space of translation bounded complex measures on G by M ∞ (G). For most of this paper, we consider M ∞ (G) equipped with the vague topology.…”

Section: General Setup and Class Of Measuresmentioning

confidence: 99%

See 1 more Smart Citation

Weighted Dirac combs with pure point diffraction

Baake¹,

Moody²

2004

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

140

374

View full text Add to dashboard Cite

show abstract

“…The measure γ m is called autocorrelation measure. Its Fourier transform exists and is called diffraction measure (see [14,15] for definition and background on Fourier transforms on measures). This measure describes the outcome of actual diffraction experiments [11,20].…”

Section: Diffraction Theorymentioning

confidence: 99%

Pure point diffraction and cut and project schemes for measures: the smooth case

2006

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Abstract. We present cut and project formalism based on measures and continuous weight functions of sufficiently fast decay. The emerging measures are strongly almost periodic. The corresponding dynamical systems are compact groups and homomorphic images of the underlying torus. In particular, they are strictly ergodic with pure point spectrum and continuous eigenfunctions. Their diffraction can be calculated explicitly. Our results cover and extend corresponding earlier results on dense Dirac combs and continuous weight functions with compact support. They also mark a clear difference in terms of factor maps between the case of continuous and non-continuous weight functions.

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“…In this setting, we call a measure transformable when it is tempered and when its Fourier transform (as a distribution) is again a measure. Transformability of a measure is a difficult question in general; see [26] and references therein for details.…”

Section: Some Recollections From Fourier Analysis and Diffraction Theorymentioning

confidence: 99%

Diffraction of Stochastic Point Sets: Explicitly Computable Examples

Baake

Birkner

Moody

2009

Commun. Math. Phys.

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Abstract. Stochastic point processes relevant to the theory of long-range aperiodic order are considered that display diffraction spectra of mixed type, with special emphasis on explicitly computable cases together with a unified approach of reasonable generality. The latter is based on the classical theory of point processes and the Palm distribution. Several pairs of autocorrelation and diffraction measures are discussed which show a duality structure analogous to that of the Poisson summation formula for lattice Dirac combs.

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Almost periodic measures

Cited by 68 publications

References 0 publications

Weighted Dirac combs with pure point diffraction

Weighted Dirac combs with pure point diffraction

Pure point diffraction and cut and project schemes for measures: the smooth case

Diffraction of Stochastic Point Sets: Explicitly Computable Examples

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