Exact reconstruction of the nonnegative measures using model sets

Abstract

In this paper we are concerned with the reconstruction of a class of measures on the square from the sampling of its Fourier coefficients on some sparse set of points. We show that the exact reconstruction of a weighted Dirac sum measure is still possible when one knows a finite number of non-adaptive linear measurements of the spectrum. Surprisingly, these measurements are defined on a model set, i.e. quasicrystal.

Keywords

Fourier analysis irregular sampling theory

1. Introduction

1.1. Background

In image processing, the problem of the exact reconstruction of a positive measure appears in several applications as image compression, superresolution problem and image denoising. The pioneering basis pursuit algorithm is used for the exact reconstruction of sparse finite dimensional vectors. The basis pursuit was introduced to the statistics community by Chen, Donoho and Saunders [4] and by earlier works of Donoho and Stark [6]. P. Doukhan, E. Gassiat and P. Gamboa considered in [8] and in [7] the exact reconstruction of a nonnegative measure in relation with superresolution problem. More recently, Y. de Castro and P. Gamboa in [5] considered the exact reconstruction of a signed measures in one dimensional case. In their paper, the main result show that the exact reconstruction of a signed measure from few values of a finite number of non-adaptive linear measurements, is possible by using the method of Beurling [1] Beurling Minimal Extrapolation.

In contrast, the results here are intrinsically multidimensional and use the arithmetical properties of simple quasicrystals defined by a cut and project scheme [16,18]. The results of the present papers has deep connections with the remarkable theory called compressed sensing. The main result of the paper can be viewed as a simplest version of the deterministic compressed sensing. Indeed, our measures are defined in a deterministic way by using quasicrystals. In the compressed sensing theory, the target function is obtained through a variational minimization procedure [2,3]. Following the same principle, in this paper we develop a related program that recovers all the nonnegative measures with small supports by using an irregular sampling in the Fourier domain defined by a simple quasicrystals.

The result of this work is related with the recently discovered sampling properties of simple quasicrystals [14,15]. The proofs of the main results are different from those used in [14] or [15]. The theory of quasicrystals introduced and develop by Y. Meyer in [16,18], was the object of several researches of J. Lagarias [10] and R. Moody which relate the arithmetical properties of the quasicrystals Λ to the analytical properties of the corresponding measure $σ_{Λ} = \sum_{λ \in Λ} δ_{λ}$ , see e.g. [10,11,21–23] and [20]. In the same spirit, one should mention the results of Nir and Olevskii on the Poisson formula and the quasicrystals in [13].

The result of the present paper give a convenient recipe to the exact reconstruction problem of the nonnegative measures by using linear non-adaptive measurements defined by simple quasicrystals. It is important to mention that there exists other type of quasicrystals, e.g. [9] for which the present result is no valid. This is a first open problem, to obtain similar results on the exact reconstruction by using general quasicrystals. Another point important to mention is that there exists mainly three tentative to define quasicrystals based on diophantine approximation [17], cut and project scheme [23] and almost lattices [11] and that these definitions are not equivalent [19]. Here we use simple quasicrystals defined by a cut and project scheme. A second open problem is to find algorithms for exact reconstruction which apply to other types of set of points.

1.2. Problem and solution

Let us explain more precisely what is done in this paper. The action takes place on $Q = {[0, 1]}^{2}$ identified to $T^{2} = {(R / Z)}^{2}$ . In image processing we often consider that the image is defined on Q an then extended by periodicity. For each integer $N > 0$ , let $F = {x_{1}, x_{2}, \dots, x_{N}}$ be a set of points in the square Q. We consider $M_{N}$ the class of measures on Q defined as follows: $ν \in M_{N}$ if $ν = \sum_{j = 1}^{N} ω_{j} δ_{x_{j}}$ , where the weights $ω_{j} ⩾ 0$ , $x_{j} \in F$ for all $1 ⩽ j ⩽ N$ and δ is the Dirac impulse. Note that the definition of a measure ν in the class $M_{N}$ depends on parameter N, on the weights $w_{j}$ , $1 ⩽ j ⩽ N$ , and on the set of points F which are considered arbitrary.

Since the unit square Q has been identified to $T^{2} = {(R / Z)}^{2}$ , the Fourier transform of an arbitrary measure μ on Q is the sequence of its Fourier coefficients defined by $\begin{matrix} (1.1) & \hat{μ} (k) = \int_{Q} exp (- 2 π i k \cdot x) d μ (x), k \in Z^{2} . \end{matrix}$

We are concerned here with the reconstruction of $ν \in M_{N}$ from the observation of the data $a (λ) = {\hat{ν} (λ), λ \in Λ}$ , i.e. from its Fourier coefficients on some sampling set $Λ \subset Z^{2}$ . More precisely, in what follows we prove that for every $α \in (0, 1 / 2)$ there exists a sparse set $Λ_{α}$ such that (a) density $Λ_{α} = 2 α$ and (b) the “irregular sampling” $\hat{ν} (λ) = a (λ)$ , $λ \in Λ_{α}$ on $Λ_{α}$ allows to recovery exactly any positive measure $ν \in M_{N}$ .

The construction of the sparse set $Λ_{α}$ follows the theory of “model sets” ([18] and [16]) and is given in next section. Let us precise the strategy of the proof. The first step is the unicity. By using the peculiar structure of the sampling set $Λ_{α}$ we show the following result: if $ν \in M_{N}$ and μ is an arbitrary measure on $T^{2}$ such that $\hat{ν} = \hat{μ}$ on the model set $Λ_{α}$ , then $ν = μ$ over $T^{2}$ . In the second step we show that $ν \in M_{N}$ has minimum mass among all the measures μ over $T^{2}$ . In other words, if we a priori know that the data $a (λ)$ , $λ \in Λ_{α}$ , are the Fourier coefficients of some nonnegative measure $ν \in M_{N}$ , then ν is the unique nonnegative solution of the following problem: $\begin{matrix} (1.2) & arg min {‖ μ ‖; μ ⩾ 0, \hat{μ} (λ) = \hat{ν} (λ), λ \in Λ_{α}}, \end{matrix}$ where $‖ μ ‖$ is the total mass of the measure μ. Note we do not impose $μ \in M_{N}$ in (1.2). The unknowns of our problem are the set of points F and the weights $w_{j}$ , $1 ⩽ j ⩽ N$ . Since $‖ μ ‖ = \hat{μ} (0)$ the problem (1.2) is the simplest version of the minimization problem used in compressed sensing.

1.3. Outline

This paper is organized in the following way. Section 2 gives a convenient recipe for constructing the sparse set Λ. Section 3 formulates the exact reconstruction of non-negative measures. It should be pointed out that the proofs in this section are different from those obtained in [14]. In Section 4 we provide several counter-examples, that prove that our results are sharp. The construction of the counter-examples follow along the lines of [14]. Finally, we have an Appendix, where we prove two lemmas used in Section 3.

2. Construction of the sampling set

In this section we construct the sampling set $Λ_{α}$ and give some properties of these sets. The recipe for constructing follows from the “model sets” theory developed by Y. Meyer (see [16]). Define $Π : R \mapsto T = R / Z$ by $Π (t) = t (mod 1)$ . Then $Z^{2}$ can be embedded in $T$ by the mapping $γ^{*} : Z^{2} \mapsto T$ defined by $\begin{matrix} (2.1) & γ^{*} (m, n) = Π (m \sqrt{2} + n \sqrt{3}) . \end{matrix}$ This mapping $γ^{*}$ is injective with a dense range. With an obvious abuse of notation we still denote by Π the canonical mapping from $R^{2}$ to $T^{2}$ . Then the dual mapping $γ : Z \mapsto T^{2}$ is defined by $γ (k) = Π (k \sqrt{2}, k \sqrt{3})$ and its range will be denoted by Γ. Then Γ is dense in $T^{2}$ .

Let $I \subset T$ an arbitrary interval (or arc) of the circle. This arc is not necessarily centered in 0 and its complement in $T$ is also an interval. Define the subset $Λ_{I} \subset Z^{2}$ by letting $\begin{matrix} (2.2) & Λ_{I} = {(p, q) \in Z^{2}; γ^{*} (p, q) \in I} . \end{matrix}$ If $I = [a, b]$ where $0 < a < b < 1$ , then $Λ_{I} = {(p, q) \in Z^{2}; \exists r \in Z, a ⩽ p \sqrt{2} + q \sqrt{3} - r ⩽ b}$ . If $I = [- α, α]$ we write $Λ_{α}$ instead of $Λ_{I}$ .

Note that the set $Λ_{α}$ is a “model set” following the terminology introduced by Y. Meyer, in [16] and [18]. In [10] more details on the properties of these sets are obtained. We know from the theory of “model sets” that the density of $Λ_{I}$ is uniform and equals $| I |$ . This notion is defined now, following [12].

Definition 2.1.
The upper density $D^{+} (Λ)$ of Λ is defined as ${lim sup}_{R \to \infty} {sup}_{x \in R^{2}} \frac{# {B (x, R) \cap Λ}}{π R^{2}}$ and the lower density $D_{-} (Λ)$ is defined by ${lim inf}_{R \to \infty} {inf}_{x \in R^{2}} \frac{# {B (x, R) \cap Λ}}{π R^{2}}$ . If $D^{+} (Λ) = D_{-} (Λ)$ then the density is called uniform.

The density of $Λ_{α} \subset Z^{2}$ is uniform and equals $2 α$ . It means that for every $ε > 0$ there exists a $R (ε)$ such that for $R ⩾ R (ε)$ and uniformly in $x_{0} \in Z^{2}$ $\begin{matrix} (2.3) & (2 α - ε) π R^{2} ⩽ card {k \in Z^{2}; Λ_{α} \cap B (x_{0}, R)} ⩽ (2 α + ε) π R^{2} . \end{matrix}$ Here $B (x 0, R)$ is the disc centered at $x_{0}$ and radius R. Note that the choice of $\sqrt{2}$ and $\sqrt{3}$ is irrelevant and other irrational numbers $ξ_{1}$ and $ξ_{2}$ could be used as well as long as $ξ_{1}$ , $ξ_{2}$ and 1 are linearly independent over $Q$ .
3. Main results

Let α be a fixed constant in $(0, 1 / 2)$ , and $Λ_{α} \subset Z^{2}$ be the model set defined by using $I = [- α, α]$ . In other words $\begin{matrix} (3.1) & Λ_{α} = {(p, q) \in Z^{2}; \exists r \in Z, | p \sqrt{2} + q \sqrt{3} - r | ⩽ α} . \end{matrix}$ Let ν be a measure in $M_{N}$ . Then the measure ν is a sum of atomic masses $ω_{j} ⩾ 0$ on the points $F = {x_{1}, x_{2}, \dots, x_{N}}$ . The parameter $N > 0$ , the weights $w_{j}$ , $1 ⩽ j ⩽ N$ and F are arbitrary. We begin our study by proving the following result:

Theorem 3.1.
Let ν be the measure defined above and $μ ⩾ 0$ be a positive measure on the torus $T^{2}$ such that $\hat{μ} (λ) = \hat{ν} (λ)$ , $λ \in Λ_{α}$ . Then $μ = ν$ .
Proof.
We define $\begin{matrix} (3.2) & ρ = μ - ν . \end{matrix}$ By definition $\hat{ρ} (λ) = 0$ , $λ \in Λ_{α}$ . We will show that $ρ = 0$ over $T^{2}$ . To begin with we prove the following lemma: Lemma 3.1.
Let θ be the triangle function on $T = R / Z$ supported on $[- α, α]$ defined by $θ (0) = 1$ , $θ (α) = θ (- α) = 0$ , θ being affine on $[α, 0]$ and $[- α, 0]$ . Then $\begin{matrix} θ (x) = \sum_{k = - \infty}^{\infty} p_{k} exp (2 π i k x), where p_{k} ⩾ 0 . \end{matrix}$ Let τ be the measure $\sum_{k = - \infty}^{\infty} p_{k} δ_{y_{k}}$ , where $y_{k} = (k \sqrt{2}, k \sqrt{3}) mod Z^{2}$ . Then $\begin{matrix} τ * ρ = 0 . \end{matrix}$
Proof.
By the definition of the measure τ, for all $(p, q) \in Z^{2}$ , we have $\begin{matrix} \hat{τ} (p, q) = \sum_{k = - \infty}^{\infty} p_{k} exp (2 π i (p \sqrt{2} + q \sqrt{3}) k) . \end{matrix}$ Therefore $\begin{matrix} \hat{τ} (p, q) = θ (p \sqrt{2} + q \sqrt{3}) . \end{matrix}$ Note that $p \sqrt{2} + q \sqrt{3} \in J = T ∖ I$ for all $(p, q) \notin Λ_{α}$ . Since θ vanishes on J it follows that $\hat{τ} (p, q) = 0$ whenever $(p, q) \notin Λ_{α}$ . Now $\hat{ρ} (λ) = 0$ whenever $λ \in Λ_{α}$ . Then $\hat{τ} \cdot \hat{ρ} = 0$ , which implies $τ * ρ = 0$ . This ends the proof of Lemma 3.1. □

Lemma 3.1 shows that $τ * (μ - ν) = 0$ over $T^{2}$ . More precisely $\begin{matrix} (3.3) & (τ * ρ) (x) = \sum_{k = - \infty}^{\infty} p_{k} (μ - ν) (x - y_{k}) = 0 . \end{matrix}$ Let us consider the measure $\tilde{μ} = τ * μ$ . By definition, the measure $\tilde{μ}$ satisfies the following identity $\begin{matrix} (3.4) & \tilde{μ} (x) = \sum_{k = - \infty}^{\infty} p_{k} (μ) (x - y_{k}) = \sum_{k = - \infty}^{\infty} \sum_{j = 1}^{N} p_{k} ω_{j} δ_{y_{k} + x_{j}} . \end{matrix}$ Note that the right hand side of (3.4) is an atomic measure supported by $Γ + F$ , where $Γ = {(k \sqrt{2}, k \sqrt{3}); k \in Z}$ and $F = {x_{1}, x_{2}, \dots, x_{N}}$ .

The set of points $Γ + F$ may be written as $Γ_{1} \cup \dots \cup Γ_{m}$ where $Γ_{j} = Γ + y_{j}$ , and $Γ_{j}$ are disjoints sets of points for all $1 ⩽ j ⩽ m$ , by using a relabeling of F if necessary.

It follows that the measure μ is absolutely continuous with respect to the measure $\tilde{μ}$ . Indeed, we have $μ ⩾ 0$ , $\tilde{μ} ⩾ 0$ and $\tilde{μ} ⩾ p_{0} μ$ . This follows directly from the definition of $\tilde{μ}$ , the term $p_{0} μ (x)$ is one of the terms in the definition of $\tilde{μ} (x)$ .

Consequently, μ is also an atomic measure supported on the set of points $Γ + F$ . Finally our measure $ρ = μ - ν$ is an atomic measure supported on $Γ + F$ . We decompose now the measure ρ as follows $\begin{matrix} ρ = ρ_{1} + \dots + ρ_{m}, where supp (ρ_{j}) \subset Γ + y_{j}, 1 ⩽ j ⩽ m . \end{matrix}$ It follows that $\begin{matrix} τ * ρ = τ * ρ_{1} + \dots + τ * ρ_{m} \end{matrix}$ and the support of each part $τ * ρ_{j}$ satisfies $supp (τ * ρ_{j}) \subset Γ + y_{j}$ , for all $1 ⩽ j ⩽ m$ .

Since $Γ_{j}$ are disjoints sets of points, by using the identity $τ * ρ = 0$ we obtain that $τ * ρ_{j} = 0$ for all $1 ⩽ j ⩽ m$ . Let us consider $μ_{j} (x - y_{j}) = ρ_{j} (x)$ . It follows that $τ * μ_{j} = 0$ , for all $1 ⩽ j ⩽ m$ . Then, $μ_{j}$ is a measure supported on Γ and $μ_{j}$ is the sum of atomic masses supported on Γ. Note that by definition the measure $ρ_{j}$ is the restriction of the whole measure $ρ = μ - ν$ to $Γ_{j}$ . From this remark, one conclude that for all $1 ⩽ j ⩽ m$ , all the weights in the definition of the measure $μ_{j}$ are positive, excepting a finite number of them. Now we need the following lemma: Lemma 3.2.
Let σ be an atomic measure supported on Γ. We assume that, excepting a finite number, all the weights in the definition of σ are nonnegatives and also we assume that $τ * σ = 0$ . Then $σ = 0$ .
Proof.
For the proof of this lemma, let us consider $σ = \sum_{k = - \infty}^{\infty} a_{k} δ_{y_{k}}$ and also $\begin{matrix} (3.5) & g (x) = \sum_{k = - \infty}^{\infty} a_{k} exp 2 π i k x . \end{matrix}$ The hypothesis $σ * τ = 0$ is equivalent to $\hat{σ} \cdot \hat{τ} = 0$ . Therefore $\begin{matrix} \hat{σ} (k, l) = 0 if k \sqrt{2} + l \sqrt{3} \in I = [- α, α] . \end{matrix}$ It follows that $g = 0$ over $I = [- α, α]$ . Let φ be a compactly supported function such that $φ \in C_{0}^{\infty} (T)$ and $supp (φ) \subset [- \frac{α}{8}, \frac{α}{8}]$ , $φ > 0$ over $] - \frac{α}{8}, \frac{α}{8} [$ . We suppose that φ is an even function. We define $Φ = φ * φ$ . It follows that $\hat{Φ} (k) ⩾ 0$ . By Lemma A.1, in Appendix, we get a more precise result, namely $\hat{Φ} (k) > 0$ , $k \in Z$ . We replace g by $G = g * Φ$ and we get $\begin{matrix} (3.6) & G (x) = \sum_{k = - \infty}^{\infty} α_{k} exp 2 π i k x, where α_{k} = a_{k} \hat{Φ} (k) . \end{matrix}$ Now, $G \in C_{0}^{\infty} (T)$ , excepting a finite number all $α_{k} ⩾ 0$ . Moreover G vanishes over $[- \frac{α}{4}, \frac{α}{4}]$ by construction. By using Lemma A.2 in Appendix, we get $G = 0$ . The proof of $σ = 0$ is done. □

The previous lemma, used for $σ = ρ$ implies that $ρ = μ - ν = 0$ , which is our claim. The proof of Theorem 3.1 is complete. □

The following theorem shows that the measure ν is the unique solution of the problem (1.2).
Theorem 3.2.
With ν as above, all measure $μ \neq ν$ (nonnecessary positive) satisfying $\hat{μ} (λ) = \hat{ν} (λ)$ , $λ \in Λ_{α}$ verify $‖ μ ‖ > ‖ ν ‖$ .
Proof.
In the case $ν ⩾ 0$ and μ a signed measure, there exist an infinity measures satisfying (1.2). We first establish $‖ μ ‖ ⩾ ‖ ν ‖$ for all these measures. To this end, we decompose μ as $μ = u - v$ with $u ⩾ 0$ and $v ⩾ 0$ and the supports of u and v are disjoint Borel sets. By definition, we have that $\begin{matrix} \hat{μ} (0) = \int u - \int v \end{matrix}$ while $‖ μ ‖ = \int u + \int v$ . Therefore $\begin{matrix} ‖ μ ‖ = \int u + \int v ⩾ \int u - \int v = \hat{μ} (0) = \hat{ν} (0) = ‖ ν ‖, \end{matrix}$ where we have used $0 \in Λ_{α}$ .

Assume now that $‖ μ ‖ = ‖ ν ‖$ , we will show that this implies $μ = ν$ . Since $\hat{μ} (λ) = \hat{ν} (λ)$ , $λ \in Λ_{α}$ and $0 \in Λ_{α}$ , we get $\begin{matrix} (3.7) & ‖ ν ‖ = \hat{ν} (0) = \hat{μ} (0) = \int u - \int v = \int u + \int v = ‖ μ ‖ . \end{matrix}$ From (3.7) we infer that $v = 0$ . Consequently $μ = u$ is a positive measure. We are now in the hypothesis of Theorem 3.1, with $\hat{μ} (λ) = \hat{ν} (λ)$ , $λ \in Λ_{α}$ and $0 \in Λ_{α}$ , $μ ⩾ 0$ , $ν ⩾ 0$ we get $μ = ν$ . Arguing by contradiction, since $\hat{μ} (λ) = \hat{ν} (λ)$ , $λ \in Λ_{α}$ we cannot have $‖ μ ‖ = ‖ ν ‖$ , without $μ = ν$ . Then $‖ μ ‖ > ‖ ν ‖$ . □

This situation corresponds to the simplest situation in the compressed sensing problem, where we try to reconstruct a function f from some partial measurements on $\hat{f}$ , through a minimization problem. In the case of measures the minimization problem of $‖ μ ‖$ not appear since $\hat{μ} (0) = \hat{ν} (0)$ . Let us emphasize that in the minimization problem (1.2) one looks for a minimizer among all signed measures on Q. Nevertheless, the target measure is assumed to be in $M_{N}$ .
4. Counter examples

In this section, we show that the positivity of the weights in the definition of measures play a key role. These theorems follows the lines of similar results obtained in [14]. For the sake of completeness the proofs are briefly sketched below.

Theorem 4.1.

Theorem 3.2 is false if $ν = \sum_{k = 1}^{N} α_{k} δ_{x_{k}}$ and $α_{k}$ are not necessarily positives.

Proof.

Let θ be the triangle function on $T = R / Z$ supported on $[α, 1 - α]$ defined by $θ (1 / 2) = 1$ , $θ (α) = θ (1 - α) = 0$ , θ being affine on $[α, 1 / 2]$ and on $[1 / 2, 1 - α]$ . The Fourier series of θ is $\begin{matrix} θ (x) = \sum_{k = - \infty}^{\infty} {(- 1)}^{k} β_{k} exp (2 π i k x), \end{matrix}$ where $β_{k} > 0$ . Define the finite measure $\begin{matrix} ν_{N} = \sum_{| k | ⩽ N} {(- 1)}^{k} β_{k} δ_{y_{k}}, y_{k} = (k \sqrt{2}, k \sqrt{3}) mod Z^{2}, \end{matrix}$ and $\begin{matrix} ρ_{N} = \sum_{| k | > N} {(- 1)}^{k} β_{k} δ_{y_{k}}, y_{k} = (k \sqrt{2}, k \sqrt{3}) mod Z^{2} . \end{matrix}$ By construction we have ${\hat{ν}}_{N} (λ) = - {\hat{ρ}}_{N} (λ)$ , $λ \in Λ_{α}$ . Note that the total mass of $ρ_{N}$ go to zero when $N \to \infty$ . It follows that for N large enough we have $‖ ρ_{N} ‖ < ‖ ν_{N} ‖$ . Consequently $ν_{N}$ is not the solution of the minimization problem $\begin{matrix} arg min {‖ μ ‖; \hat{μ} (λ) = {\hat{ν}}_{N} (λ), λ \in Λ_{α}} . \end{matrix}$ The counter-example is then obtained for the weights ${(- 1)}^{k} β_{k}$ . □

Theorem 4.2.

Let $ν ⩾ 0$ be an arbitrary positive measure not necessary of the form of measures in $M_{N}$ and μ be an arbitrary measure. Then the Theorem 3.1 is false.

Proof.

It uses the same methods as [14]. □

Footnotes

Acknowledgements

The roots of this paper are in Meyer’s deep theory of “model sets” introduced and developed in several works, see for example [16,]. The author gratefully acknowledges Yves Meyer for its constant support and helpful discussions on the subject. The author acknowledges also John Benedetto and Joaquim Ortega-Cerdà for theirs encouragement.

We establish here the following lemmas used in Section 3.

References

[1]

Beurling, Collected Works of Arne Beurling (2 Vols),

Carleson et al., eds, Birkhauser, 1989.

[2]

E.J.

Candès,

J.K.

Romberg and

Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inform. Theory 52(2) (2006), 489–509.

[3]

E.J.

Candès,

J.K.

Romberg and

Tao, Stable signal recovery from incomplete and inaccurate measurements, Comm. Pure Appl. Math. 59(8) (2006), 1207–1223.

[4]

S.S.

Chen,

D.L.

Donoho and

M.A.

Saunders, Atomic decomposition by basis pursuit, SIAM J. Sci. Comput. 20(1) (1998), 33–61.

[5]

De Castro and

Gamboa, Exact reconstruction using Beurling minimal extrapolation, Journal of Mathematical Analysis and Applications 395(1) (2012), 336–354.

[6]

D.L.

Donoho and

P.B.

Stark, Uncertainty principles and signal recovery, SIAM J. Appl. Math. 49(3) (1989), 906–931.

[7]

Doukhan and

Gamboa, Superresolution rates in Prokhorov metric, Canad. J. Math. 48(2) (1996), 316–329.

[8]

Gamboa and

Gassiat, Sets of superresolution and the maximum entropy method on the mean, SIAM J. Math. Anal. 27(4) (1996), 1129–1152.

[9]

J.-B.

Gouéré, Quasicrystals and almost-periodicity, Comm. Math. Phys. 255 (2005), 655–681.

10.

[10]

J.C.

Lagarias, Meyer’s concept of regular model and quasiregular sets, Comm. Math. Phys. 179 (1996), 365–376.

11.

[11]

J.C.

Lagarias, Geometric models for quasicrystals I. Delone sets of finite type, Discrete & Computational Geometry 21 (1999), 161–191.

12.

[12]

H.J.

Landau, Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math. 117 (1967), 37–52.

13.

[13]

Lev and

Olevskii, Measures with uniformly discrete support and spectrum, C. R. Math. Acad. Sci. Paris 351 (2013), 613–617.

14.

[14]

Matei and

Meyer, A variant of compressed sensing, Revista Matematica Iberoamericana 25 (2009), 669–692.

15.

[15]

Matei and

Meyer, Quasicrystals are sets of stable sampling, Complex Variables and Elliptic Equations 55 (2010), 947–964.

16.

[16]

Meyer, Nombres de Pisot, Nombres de Salem et Analyse Harmonique, Lecture Notes in Mathematics, Vol. 117, 1970.

17.

[17]

Meyer, Algebraic Numbers and Harmonic Analysis, North-Holland, 1972.

18.

[18]

Meyer, Trois problèmes sur les sommes trigonométriques, Astérisque, Vol. 1, SMF, 1973.

19.

[19]

Meyer, Quasicrystals, almost periodic patterns, mean periodic functions and irregular sampling, African Diaspora Journal of Mathematics 13(1) (2012), 1–45.

20.

[20]

R.V.

Moody, Meyer sets and their duals, in: Proceedings of the NATO Advanced Study Institute on Long-Range Aperiodic Order,

R.V.

Moody, ed., NATO ASI Series, Vol. C489, Kluwer Acad. Press, 1997, pp. 403–441.

21.

[21]

R.V.

Moody, Model sets: A survey, in: From Quasicrystals to More Complex Systems,

Axel,

Dénoyer and

J.P.

Gazeau, eds, Centre de physique Les Houches, Springer-Verlag, 2000.

22.

[22]

R.V.

Moody, Uniform distribution in model sets, Canad. Math. Bull. 45(1) (2002), 123–130.

23.

[23]

R.V.

Moody, Mathematical quasicrystals: A tale of two topologies, in: Proceedings of the International Congress of Mathematical Physics, Lisbon, 2003.