On the Fourier Transformability of Strongly Almost Periodic Measures

Strungaru, Nicolae

doi:10.4153/s0008414x19000075

Cited by 5 publications

(4 citation statements)

References 51 publications

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“…Finally, since

S

is weakly uniformly discrete, it follows that

ν

is translation bounded. Now, since the measure

μ

is tempered and its Fourier transform as a tempered distribution is a translation‐bounded measure,

μ

is Fourier transformable as a measure, by [44, Theorem 5.2]. The same statements hold for the measure

ν

, and we have

\hat{μ} = ν

as claimed.

□

…”

Section: Specific Results For G=double-struckrdmentioning

confidence: 59%

“…In previous sections, we have assumed our measures to be translation bounded and Fourier transformable. The connection between the distributional Fourier transform and Fourier transformability as an unbounded Radon measure, as we have considered, was clarified in [44], where it was shown that a measure

μ

R^{d}

is Fourier transformable as a measure if and only if it is tempered and its distributional Fourier transform is a translation‐bounded measure. Thus, in the Euclidean setting, a measure

μ

is translation bounded and Fourier transformable if and only if its distributional Fourier transform

ν

is a translation‐bounded and Fourier‐transformable measure.…”

Section: Specific Results For G=double-struckrdmentioning

confidence: 99%

“…Recalling from Section 5 that positive definite Radon measures are Fourier transformable, and again using [44, Theorem 5.2], we summarise some useful sufficient conditions as follows. Corollary Let

μ

be a tempered measure on

R^{d}

such that its distributional Fourier transform,

ν

, is a measure.…”

Section: Specific Results For G=double-struckrdmentioning

confidence: 99%

See 2 more Smart Citations

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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“…Finally, since

S

is weakly uniformly discrete, it follows that

ν

is translation bounded. Now, since the measure

μ

is tempered and its Fourier transform as a tempered distribution is a translation‐bounded measure,

μ

is Fourier transformable as a measure, by [44, Theorem 5.2]. The same statements hold for the measure

ν

, and we have

\hat{μ} = ν

as claimed.

□

…”

Section: Specific Results For G=double-struckrdmentioning

confidence: 59%

μ

R^{d}

is Fourier transformable as a measure if and only if it is tempered and its distributional Fourier transform is a translation‐bounded measure. Thus, in the Euclidean setting, a measure

μ

is translation bounded and Fourier transformable if and only if its distributional Fourier transform

ν

is a translation‐bounded and Fourier‐transformable measure.…”

Section: Specific Results For G=double-struckrdmentioning

confidence: 99%

See 1 more Smart Citation

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

Baake

Strungaru

Terauds

2020

Trans. London Math. Soc.

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“…Here, we consider the important special case of measures with uniformly discrete support, for which the three key notions turn out to be equivalent. This class is particularly relevant in the theory of aperiodic order, with several applications to mathematical quasicrystals and Meyer sets; see [4,8,11,12,14,17,18] and references therein.…”

Section: Proof Define the Non-negative Functionsmentioning

confidence: 99%

A note on tempered measures

Baake

Strungaru²

2023

Colloq. Math.

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Tempered distributions with translation bounded measure as Fourier transform and the generalized Eberlein decomposition

Spindeler,

Strungaru

2023

Mathematische Nachrichten

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In this paper, we study the class of tempered distributions whose Fourier transform is a translation bounded measure and show that each such distribution in has order at most 2d. We show the existence of the generalized Eberlein decomposition within this class of distributions, and its compatibility with all previous Eberlein decompositions. The generalized Eberlein decomposition for Fourier transformable measures and properties of its components are discussed. Lastly, we take a closer look at the absolutely continuous spectrum of measures supported on Meyer sets.

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On the Fourier Transformability of Strongly Almost Periodic Measures

Cited by 5 publications

References 51 publications

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

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

A note on tempered measures

Tempered distributions with translation bounded measure as Fourier transform and the generalized Eberlein decomposition

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