Fourier transformable measures with weak Meyer set support and their lift to the cut-and-project scheme

Abstract In this paper, we prove that given a cut-and-project scheme 
$(G, H, \mathcal {L})$
 and a compact window 
$W \subseteq H$
 , the natural projection gives a bijection between the Fourier transformable measures on 
$G \times H$
 supported inside the strip 
${\mathcal L} \cap (G \times W)$
 and the Fourier transformable measures on G supported inside 
${\LARGE \curlywedge }(W)$
 . We provide a closed formula relating the Fourier transform of the original measure and the Fourier transform of the projection. We show that this formula can be used to re-derive some known results about Fourier analysis of measures with weak Meyer set support.


Introduction
After the discovery of quasicrystals [39], it has become clear that we need to better understand the process of diffraction. Mathematically, the diffraction pattern of a solid can be viewed as the Fourier transformγ of the autocorrelation measure γ of the structure (see [13] for the setup and the monographs and see [3,4] for a general review of the theory). The measure γ is positive-definite, and therefore it is Fourier transformable as a measure [1,8,31] with positive Fourier transformγ. It is this measureγ, which models the diffraction of our solid.
The best mathematical models for Delone sets with a large pure point spectrum and (generic) positive entropy are Meyer sets. They have been introduced in the pioneering work of Meyer [26], and popularized in the area of Aperiodic Order by Moody [28,29] and Lagarias [18,19]. They are usually constructed via a cut-andproject scheme (or simply CPS) and can be characterized via harmonic analysis, discrete geometry, algebra, and almost periodicity [26,29,46]. The basic idea behind a CPS is to project points from a higher-dimensional lattice, which lie within a bounded strip of the real space, into the real space (see Definition 2.8 for the exact definition). If the cross section of the strip (called the window) is regular, then the resulting model set is pure point diffractive [7,14,35,38]. Recent work proved pure point diffractivity for a larger class of weak model sets [5,[15][16][17]48].
As subsets of regular model sets, Meyer sets still exhibit a large pure point spectrum [43][44][45][46][47]49] and a highly ordered continuous spectrum [43,45,47,49]. The long-range order of the spectrum of Meyer sets can be traced to that of a covering regular model set [47,49], and can be derived from the Poisson summation formula for the lattice in the CPS [3,25,34,35].
One would expect it to be possible to relate the diffraction of a Meyer set (or more generally a measure with Meyer set support) directly to the lattice L in the CPS. It is the goal of this paper to establish this connection. Let us briefly explain our approach.
Fix a CPS (G, H, L) and a compact set W ⊆ H. It is easy to see that establishes a bijection between translation bounded measures supported inside ⋏(W) and translation bounded measures supported inside L ∩ (G × W). We first show in Proposition 3.6 that γ is positive-definite if and only if η is positive-definite. Since each Fourier transformable measure supported inside a Meyer set can be written as a linear combination of positive-definite measures supported inside a common model set, we establish in Theorem 4.1 that γ is Fourier transformable if and only if η is Fourier transformable, and relate their Fourier transform (see (4.1)). We complete the paper by discussing in Section 5 how these results can be used to re-derive the known properties of diffraction for measures with weak Meyer set support, and potentially used to prove new results.

Definitions and notations
Throughout the paper, G denotes a second countable locally compact Abelian group (LCAG). By C u (G), we denote the space of uniformly continuous and bounded functions on G. This is a Banach space with respect to the sup norm ∥.∥ ∞ . As usual, we denote by C 0 (G) the subspace of C u (G) consisting of functions vanishing at infinity, and by C c (G) the subspace of compactly supported continuous functions. Note that In the spirit of [10], we denote by Given two LCAG's G and H and two functions g In the rest of this section, we review some of the basic concepts which are important for this paper. For a more general review of these, we recommend [3,4].

Measures
In the spirit of Bourbacki [9], by a measure, we understand a linear functional on C c (G) which is continuous with respect to the inductive topology. This notion corresponds to the classical concept of a Radon measure (see [35,Appendix]). For the case G = R d , a clear exposition of this is given in [3].
By the Riesz representation theorem [37], a positive Radon measure is simply a positive regular Borel measure. Moreover, each Radon measure is a linear combination of (at most four) positive Radon measures [35,Appendix].
Next, we review the total variation of a measure.

Definition 2.2
Given a measure μ, we can define [32,33,35] a positive measure |μ|, called the total variation of μ, such that, for all φ ∈ C c (G) with φ ≥ 0, we have We are now ready to introduce the concept of translation boundedness for measures and norm almost periodicity.

Definition 2.3
Let A ⊆ G be a fixed precompact set with nonempty interior. We define the A-norm of μ via Then (M ∞ (G), ∥.∥ A ) is a normed space. It is in fact a Banach space [36].
Next, we review the definition of norm almost periodicity as introduced in [7].

Definition 2.5 Let
A ⊆ G be a fixed precompact set with nonempty interior. A measure μ ∈ M ∞ (G) is called norm almost periodic if, for each ε > 0, the set P A ε (μ) ∶= {t ∈ G ∶ ∥T t μ − μ∥ A < ε} of ε-norm almost periods of μ is relatively dense.
As discussed above, different precompact sets define equivalent norms. This means that while the set of ε-norm almost periods on μ depends on the choice of A, the almost periodicity of μ is independent of this choice.
Any norm almost periodic measure is strongly almost periodic [7], and the two concepts are equivalent for measures with Meyer set support [7]. In general, norm almost periodicity is an uniform version of strong almost periodicity [42,Theorem 4.7]. The class of norm almost periodic pure point measure was studied in detail and characterized in [46].
Let us next recall positive-definiteness for functions and measures. For more details, we recommend [8,31].
We complete the subsection by reviewing the notion of Fourier transformability for measures. For a more detailed review of the subject, we recommend [31].

Cut-and-project schemes and Meyer sets
In this part, we review some notions related to the cut-and-project formalism. For more details, we recommend [3,28,29].

Definition 2.8
By a CPS, we understand a triple (G, H, L) consisting of a second countable LCAG G, an LCAG H, and a lattice L ⊆ G × H such that: The restriction π G | L of the first projection π G to L is one to one.
Given a CPS (G, H, L), we will denote by L ∶= π G (L). Then, π G induces a group isomorphism between L and L. Composing the inverse of this with the second projection π H , we get a mapping ⋆ ∶ L → H, which we will call the ⋆-mapping. We then have Given a CPS (G, H, L) and a subset W ⊆ H we can define When W is precompact, we will call ⋏(W) a weak model set. If furthermore W has nonempty interior ⋏(W) is called a model set.
Next, let us review the concept of a Meyer set, which plays a fundamental role in this paper.
For equivalent characterizations of Meyer sets, see [18,19,26,28,46]. Of importance to us will be the following result.

Theorem 2.10 ([46]) Let Λ ⊆ G be relatively dense. Then Λ is Meyer if and only if it is a subset of a (weak) model set.
Moreover, if Λ is Meyer, it is a subset of a weak model set in a CPS (G, H, L) with metrizable and compactly generated H.
We should emphasize here that the key for all results below is that fact that a Meyer set is a subset of a model set, and relative denseness plays no role. Because of this, in the spirit of [49], we will refer to an arbitrary subset of a (weak) model set as a weak Meyer set. It is obvious that a subset of a weak Meyer set is a weak Meyer set and that a measure is supported inside a Meyer set if and only if its support is a weak Meyer set.
Given a CPS (G, H, L), the map is a group isomorphism, and hence it induces an isomorphism between the spaces of (bounded) functions on L and L, respectively. Since L is a discrete group, the space of (translation bounded) measures on L can be identified with the space of (bounded) functions on L. On another hand, L is typically dense in G, and many functions on L do not induce pure point measures on G.
For us, of interest will be measures supported inside weak model sets ⋏(W). Since ⋏(W) is uniformly discrete [28], the space of (translation bounded) measures on ⋏(W) can be identified with the space of (bounded) functions on ⋏(W), and corresponds via the above isomorphism with the spaces of (translation bounded) measures or (bounded) functions on L, respectively, that are supported inside G × W.
Our focus in this paper is on these two spaces. We will study them as spaces of measures, and we will be interested in the relation between the Fourier theory of these two spaces, and the behavior of the Fourier transform with respect to the isomorphism induced by (2.1). For this reason, let us introduce the following notations.
Given a CPS (G, H, L) and a compact set W, we denote by Let us note here in passing that P G ,H,L,W is simply the pushforward via f −1 .
We will refer to these mappings as the lift operator and the projection operator, respectively. When the CPS and window are clear from the context, we will simply write L(μ) and P(ν), respectively, instead of L G ,H,L,W (μ) and P G ,H,L,W (ν), respectively.
The main results in this paper are that these operators are bijections between the subspaces of Fourier transformable (or cones of positive-definite) measures, and relate their Fourier transforms.
To understand the connection between the Fourier transforms, let us recall the notion of dual CPS. Given a CPS (G, H, L), we can define Then, (Ĝ,Ĥ, L 0 ) is a CPS [5,28,29,46]. We will refer to this as the CPS dual to (G, H, L).

Positive-definite measures with weak Meyer set support
In this section, we show that L G ,H,L,W and P G ,H,L,W take positive-definite measures to positive-definite measures.
Let us start with the following obvious lemma, which follows immediately from Definition 2.6 and the fact that the function from (2.1) is a group isomorphism. Let us recall now the following result of [24], which we will use often in the paper.
Then f is a positive-definite function on G Proposition 3.2. Definition 2.6 immediately gives that the restriction g = f | L to the subgroup L is a positive-definite function on L. ∎ We will also need the following result.
This induces an equivalence relation on the set {x 1 , . . . , x n }, and hence we can partition this set in equivalence classes F 1 , . . . , F m .
To make the computation clearer, define c ∶ G → C We are now ready to prove the following result.

N. Strungaru
Proof ⇒: Denote as usual L ∶= π G (L). Define g ∶ L → C via Then, by Lemma 3.3, g is a positive-definite function on L and hence, by Lemma is a positive-definite function on L. Therefore, by Lemma 3.4, the function

The lift of Fourier transformable measures
We can now prove that, given a CPS (G, H, L) and a compact set K, the lifting operator induces a bijection between the space of Fourier transformable measures supported inside ⋏(W) and the space of Fourier transformable measures supported inside L ∩ (G × W).

Proof
⇒ By [47,Lemma 8.3], there exist a compact set W ⊆ K and four positivedefinite measures ω 1 , ω 2 , ω 3 , ω 4 supported inside ⋏(K) such that Then, we have Now, by Proposition 3.6, for all 1 ≤ j ≤ 4, the measure L G ,H,L,K (ω j ) is positivedefinite. Therefore, as a linear combination of positive-definite measures, η is Fourier transformable. ⇐ . Our argument is similar to [34]. First, fix an arbitrary φ ∈ K 2 (H) so that φ ≡ 1 on W. We split the rest of the argument into two steps.
Since G is second countable, so isĜ [33]. In particular,Ĝ is σ-compact [33]. Therefore, there exists a sequence K n of compact sets with K n ⊆ (K n+1 ) ○ such that Then, ψ nφ ∈ C c (Ĝ) and by the definition of (q η)φ, we have Now, for all n, we have by (4.2) Therefore, by the dominated convergence theorem [33, Theorem 3.2.51], we havê Next, by the monotone convergence theorem [33], we have Note that for each n, we have Since q ϕ ⊗φ ∈ L 1 (|η|), we get This shows that | q ϕ| ∈ L 1 (|(η)φ|). Therefore, ψ n q ϕ is dominated by | q ϕ| ∈ L 1 (|(η)φ|) and converges pointwise to q ϕ. Thus, by (4.4) and the dominated convergence theorem, we getη Finally, by the Fourier transformability of η, we havê Therefore, we proved that for all ϕ ∈ K 2 (G), we have q ϕ ∈ L 1 (|(η)φ|) and This proves that γ is Fourier transformable and completing the proof. ∎ Using the fact that L is a bijection with inverse P, we get the following corollary.

Applications
In this section, we will discuss the relation (4.1) and how can it be used to (re)derive some results from [47].
To make the things easier to follow, we introduce the notion of strongly admissible functions.

Strongly admissible functions for CPS
Let us start with the following definition.

Definition 5.1
Given a group H of the form H = R d × H 0 , with a LCAG H 0 , a function f ∶ H → C is called strongly admissible if there exists g ∈ C u (R d ) and φ ∈ C c (H 0 ) such that: Next, given a CPS (G, H, L), we will denote by M L (G × H), the space of L-periodic measures on G × H. Note that by [21, Proposition 6.1] We will see below that given a Fourier transformable measure γ with weak Meyer set support, Theorem 4.1 can be used to create a CPS (G, H = R d × H 0 , L), an L 0periodic measure ρ(=η) and a strongly admissible function f onĤ = R d ×Ĥ 0 such that, equation This measure is strongly almost periodic by [21,Theorem 3.1]. In fact, the strong admissibility of f immediately implies that ρ f is norm almost periodic.
Indeed, let (G, H = R d × H 0 , L), let f = g ⊗ φ be strongly admissible, and let ρ ∈ M L (G × H). Pick any compact set supp(φ) ⊆ W ⊆Ĥ 0 , and let K, K 1 ⊆ G be compact sets in G with K ⊆ K ○ 1 . Then, a standard computation similar to [47,Lemma 5.2] shows that This immediately gives the following stronger version of [47, Lemma 5.2]. , and let f ∈ C 0 (H) be strongly admissible. Then, ρ f is a norm almost periodic measure.

Fourier transform of measures with weak Meyer set support
Fix an arbitrary Meyer set Λ and a Fourier transformable measure γ with supp(γ) ⊆ Λ. By Theorem 2.10 and the structure theorem of compactly generated groups, there exists a CPS (G, R d × Z m × K, L) with compact K and a compact W ⊆ R d × Z m × K such that Λ ⊆ ⋏(W) .
By eventually enlarging W, we can assume without loss of generality that It is easy to see that we can find function φ ∈ C ∞ c (R d ) ∩ K 2 (R d ) and ψ ∈ K 2 (H 0 ) with the following properties: It follows that f ∶=φ =φ ⊗ψ (5.1) is a strongly admissible function ofĤ = R d ×Ĥ 0 .

Generalized Eberlein decomposition
In this subsection, we show a pseudo-compatibility of the mapping ρ → (ρ) f of (5.3), for L periodic ρ ∈ M L (G × H) and strongly admissible f, with respect to the Lebesgue decomposition. We explain this, as well as our meaning of "pseudo-compatibility" below.
First, it is easy to see that the map satisfies: for all c 1 , . . . , c m ∈ C, ψ 1 , . . . , ψ m ∈ C c (Ĝ), ϕ 1 , . . . , ϕ m ∈ C c (Ĥ) is well defined, linear, and continuous with respect to the inductive topology. Therefore, L α can be uniquely extended to a measure P α (ρ), which is L invariant and satisfies (5.3). Now, exactly as above, let γ be a Fourier transformable measure supported inside a Meyer set Λ, and let (G, H, L), η, φ, ϕ, ψ be as in Section 5.2. Let f be as in (5.1), and let ρ =η.

Discussion
We have seen in this section that the Fourier transform of a measure γ with weak Meyer set support can be describe via (5.2) as the projection in the dual CPS of a L 0 -periodic measure via a strongly admissible function. We used this result to (re)derive properties ofγ, and we expect that this connection will lead to some new applications in the future. Indeed, while now we know quite a few properties of the Fourier transform of measures with weak Meyer set support [2,[43][44][45][46][47][48][49], we know much more about fully periodic measures in LCAG (see, for example, [36]). Moreover, the strong admissibility of f is likely to transfer many properties from ρ to ρ f . It is also worth pointing out that, while the strong admissibility of f was sufficient to derive the conclusions in this section, in fact f can be chosen of the form where g =φ ∈ S(R d ) is the Fourier transform of some φ ∈ C ∞ c (R d ); P is a trigonometric polynomial, that is, a sum of characters, that is, P = ∑ m j=1 χ j for some χ 1 , . . . , χ j ∈T m and ψ ∈ C c (K) is the characteristic function of {0}. These properties are much stronger than strong admissibility, and have the potential to lead to nice applications in the future.