Meyer Sets and Their Duals

Moody, Robert V.

doi:10.1007/978-94-015-8784-6_16

Cited by 165 publications

(206 citation statements)

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“…Such tilings have been proposed as models for quasicrystalline structures ( [2], [6], [7], [22], [24], [32]); they also arise in constructions of compactly supported wavelets and multiwavelets ( [1], [4], [5]). An alternate method of modelling quasicrystalline structures uses discrete sets, more specifically Delone sets (defined below), see [14], [15], [16], [26], [27], [28], [33]. These sets model the atomic positions occupied in the structure.…”

Section: Introductionmentioning

confidence: 99%

Substitution Delone Sets

Lagarias¹,

Yang²

2003

Discrete and Computational Geometry

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Substitution Delone set families are families of Delone sets X = (X 1 , . . . , X n ) which satisfy the inflation functional equationin which A is an expanding matrix, i.e. all of the eigenvalues of A fall outside the unit circle. Here the D ij are finite sets of vectors in R d and denotes union that counts multiplicity.This paper characterizes families X = (X 1 , ..., X n ) that satisfy an inflation functional equation, in which each X i is a multiset (set with multiplicity) whose underlying set is discrete. It then studies the subclass of Delone set solutions, and gives necessary conditions on the coefficients of the inflation functional equation for such solutions X to exist. It relates Delone set solutions to a narrower subclass of solutions, called self-replicating multi-tiling sets, which arise as tiling sets for self-replicating multi-tilings.

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Section: Introductionmentioning

confidence: 99%

Substitution Delone Sets

Lagarias¹,

Yang²

2003

Discrete and Computational Geometry

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“…Note that such measures are the main object in the theory of Fourier quasicrystals (see [1]- [12]). The following result is valid:…”

Section: Denote By S(rmentioning

confidence: 99%

Some properties of measures with discrete support

Favorov¹

2016

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. Some properties of measures with discrete support, Mat. Stud. 46 (2016), 189-195. We give some new conditions for the support of a discrete measure on Euclidean space to be a finite union of translated lattices. In particular, we consider the case when values of masses a λ of discrete measure satisfy the equality G(a λ ,ā λ ) = 0 for each analytic function G(z, w). Denote by S(R. These norms generate topology on S(R d ), and elements of the spaceMoreover, this estimate is sufficient for distribution f to be in S ′ (R d ) (see [16], Ch.3). The Fourier transform of a tempered distribution f is defined by the equalitŷis the Fourier transform of the function φ. Note that the Fourier transform of each tempered distribution is also a tempered distribution.In the paper we consider only the case when f is a measure µ on R d . We say that µ is translation bounded, if its variations on balls of radius 1 are uniformly bounded. If the Fourier transformμ is an atomic measure, then spectrum of µ is the set, and by δ λ the unit mass at the point λ. For a measure µ denote by |µ|(t) the value of its variation on the ball B(t), and by |µ| the value of its total variation, if it is finite. A measure µ is slowly increasing, if |µ|(t) grows at most polynomially as t → ∞.

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“…The vertices or the centers, in the sense of Voronoi cells, of the tiles are projections of subsets of the lattice. A detailed description of the underlying theory can be found in [16]. A large group of deterministic tilings can be constructed this way, for instance the Penrose Tiling (cf.…”

Section: Examplesmentioning

confidence: 99%