The Inverse Problem of Pure Point Diffraction—Examples and Open Questions

Terauds, Venta

doi:10.1007/s10955-013-0790-0

Cited by 9 publications

(13 citation statements)

References 9 publications

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“…For the next example the existence of the autocorrelation was established in [17]. cos(2π(4 n − 1)(·)) 4 n δ Z 2.4 n , whose Fourier transform is the measure ω := δ 2Z + n≥1 δ 2.4 n Z * (δ 4 n −1 + δ 1−4 n ) .…”

Section: Tempered Distributions With An Autocorrelationmentioning

confidence: 97%

Diffraction Theory and Almost Periodic Distributions

Strungaru¹,

Terauds²

2016

J Stat Phys

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Abstract. We introduce and study the notions of translation bounded tempered distributions, and autocorrelation for a tempered distrubution. We further introduce the spaces of weakly, strongly and null weakly almost periodic tempered distributions and show that for weakly almost periodic tempered distributions the Eberlein decomposition holds. For translation bounded measures all these notions coincide with the classical ones. We show that tempered distributions with measure Fourier transform are weakly almost periodic and that for this class, the Eberlein decomposition is exactly the Fourier dual of the Lesbegue decomposition, with the Fourier-Bohr coefficients specifying the pure point part of the Fourier transform. We complete the project by looking at few interesting examples.

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Section: Tempered Distributions With An Autocorrelationmentioning

confidence: 97%

Diffraction Theory and Almost Periodic Distributions

Strungaru¹,

Terauds²

2016

J Stat Phys

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“…10 and 11] as well as [4,15] for additional examples, and [20,15] for connections with the dynamical spectrum. Nevertheless, as is apparent from a comparison with the pure point diffraction case [23,12,58,77,76], the status of general results is lagging behind. Even for many important examples, some of the most obvious questions are still open from a mathematical point of view.…”

Section: Discussionmentioning

confidence: 99%

Non-Periodic Systems with Continuous Diffraction Measures

Baake

Birkner

Grimm

2015

Mathematics of Aperiodic Order

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Abstract. The present state of mathematical diffraction theory for systems with continuous spectral components is reviewed and extended. We begin with a discussion of various characteristic examples with singular or absolutely continuous diffraction, and then continue with a more general exposition of a systematic approach via stationary stochastic point processes. Here, the intensity measure of the Palm measure takes the role of the autocorrelation measure in the traditional approach. We furthermore introduce a 'Palm-type' measure for general complex-valued random measures that are stationary and ergodic, and relate its intensity measure to the autocorrelation measure.

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“…From the mathematical point of view, the relation between diffraction spectra and dynamical spectra of the associated dynamical system (under translation action) is now much better understood; see [11] and references therein for recent developments. Also, with methods from the theory of (stochastic) point processes, the inverse problem of structure determination from a pure point (or Bragg) diffraction spectrum has been understood in an abstract setting in rather large generality [19]; see also [23,24] for additional examples in this context. The corresponding problem for general (mixed) spectrum is still open, and appears to be rather challenging.…”

Section: Discussionmentioning

confidence: 99%