On the Fourier analysis of measures with Meyer set support

Strungaru, Nicolae

doi:10.1016/j.jfa.2019.108404

Cited by 16 publications

(24 citation statements)

References 53 publications

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“…Hence, in this article, we discuss deformed weighted model sets as a substantial subclass of SAP measures. We will prove that deformed weighted model sets with continuous and compactly supported weight and deformation functions are strongly almost periodic, thereby extending older results for weighted model sets [11,45,58,65,66]. We will then deform such structures using continuous almost periodic modulations.…”

Section: Introductionmentioning

confidence: 60%

Modulated crystals and almost periodic measures

et al. 2020

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Modulated 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-analyze these structures using methods from modern mathematical diffraction theory, thereby providing a coherent view over that class. Similar to de Bruijn’s analysis, we find stability with respect to almost periodic modulations.

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Section: Introductionmentioning

confidence: 60%

Modulated crystals and almost periodic measures

et al. 2020

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“…The matrix B is obtained by first taking the ?-map of the setvalued displacement matrix T of equation 4and then its inverse Fourier transform. For this reason, B is called the internal Fourier matrix (Baake & Grimm, 2019b), to distinguish it from the Fourier matrix of the renormalization approach in physical space (Baake & Gä hler, 2016;Baake, Frank et al, 2019); see Bufetov & Solomyak (2018, 2020 for various extensions with more flexibility in the choice of the interval lengths. In Dirac notation, we set jhi ¼ ðh a ; h b Þ T , which satisfies jhð0Þi ¼ jvi with the right eigenvector jvi of the substitution matrix M from equation 2.…”

Section: Renormalization and Internal Cocyclementioning

confidence: 99%

“…In particular, these matrices satisfy B ð1Þ ¼ B and B ðnÞ ð0Þ ¼ M n for all n 2 N, where M is the substitution matrix from equation 1, as well as the relations B ðnþmÞ ðyÞ ¼ B ðnÞ ðyÞ B ðmÞ ð n yÞ ð 19Þ for any m; n 2 N. Note that B ðnÞ ðyÞ defines a matrix cocycle, called the internal cocycle, which is related to the usual inflation cocycle (in physical space) by an application of the ?-map to the displacement matrices of the powers of the inflation rule; compare Baake, Gä hler & Mañ ibo (2019), Baake & Grimm (2019b), and see Bufetov & Solomyak (2018, 2020 for a similar approach. Note also that jj < 1, which means that n approaches 0 exponentially fast as n !…”

Section: Renormalization and Internal Cocyclementioning

confidence: 99%

“…The result on the pure point nature of diffraction holds for rather general setups, including cut-and-project schemes with non-Euclidean internal spaces. It has recently been generalized to weak model sets of extremal densities (Baake et al, 2017; Richard & Strungaru, 2017b), for which the window may even entirely consist of boundary, that is, has no interior; see also Strungaru (2017Strungaru ( , 2020 for recent work on pure point spectra.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Inflation versus projection sets in aperiodic systems: the role of the window in averaging and diffraction

Baake¹,

Grimm²

2020

Acta Cryst Sect A

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Tilings based on the cut-and-project method are key model systems for the description of aperiodic solids. Typically, quantities of interest in crystallography involve averaging over large patches, and are well defined only in the infinite-volume limit. In particular, this is the case for autocorrelation and diffraction measures. For cut-and-project systems, the averaging can conveniently be transferred to internal space, which means dealing with the corresponding windows. In this topical review, this is illustrated by the example of averaged shelling numbers for the Fibonacci tiling, and the standard approach to the diffraction for this example is recapitulated. Further, recent developments are discussed for cut-and-project structures with an inflation symmetry, which are based on an internal counterpart of the renormalization cocycle. Finally, a brief review is given of the notion of hyperuniformity, which has recently gained popularity, and its application to aperiodic structures.

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“…Without loss of generality, we may assume that

F

and

F^{'}

are minimal in this sense. Then, applying [45, Remark 5] to the measure

γ = \hat{μ}

, we gain the existence of a finite measure

ν

G

such that

μ^{†} = \hat{\overset{μ}{true}} = (\sum_{x \in normalΓ} \sum_{χ \in F^{'}} χ (x) δ_{x}) * ν,

where

μ^{†} false(g false) : = μ false(g \circ I false)

with

I (x) = - x

, so

{(μ^{†})}^{†} = μ

and

{(μ * ν)}^{†} = μ^{†} * ν^{†}

. Consequently, with

{(δ_{x})}^{†} = δ_{- x}

and

χ false(- x false) = \bar{χ (x)}

, one gets

μ = (\sum_{x \in normalΓ} \sum_{χ \in F^{'}} \bar{χ false(x false)} δ_{x}) * ν^{†} .

Define the measures

ν_{1} : = \sum_{x \in Γ + F} ν^{†} (false{x false})

…”

Section: Measures With Meyer Set Support and Sparse Fbsmentioning

confidence: 99%

Pure point measures with sparse support and sparse Fourier–Bohr support

Baake

Strungaru

Terauds

2020

Trans. London Math. Soc.

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Fourier transformable Radon measures are called doubly sparse when both the measure and its transform are pure point measures with sparse support. Their structure is reasonably well understood in Euclidean space, based on the use of tempered distributions. Here, we extend the theory to second countable, locally compact Abelian groups, where we can employ general cut and project schemes and the structure of weighted model combs, along with the theory of almost periodic measures. In particular, for measures with Meyer set support, we characterise sparseness of the Fourier-Bohr spectrum via conditions of crystallographic type, and derive representations of the measures in terms of trigonometric polynomials. More generally, we analyse positive definite, doubly sparse measures in a natural cut and project setting, which results in a Poisson summation type formula.

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On the Fourier analysis of measures with Meyer set support

Cited by 16 publications

References 53 publications

Modulated crystals and almost periodic measures

Modulated crystals and almost periodic measures

Inflation versus projection sets in aperiodic systems: the role of the window in averaging and diffraction

Pure point measures with sparse support and sparse Fourier–Bohr support

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