Aperiodic order and spherical diffraction, III: The shadow transform and the diffraction formula

We define spherical diffraction measures for a wide class of weighted point sets in commutative spaces, i.e. proper homogeneous spaces associated with Gelfand pairs. In the case of the hyperbolic plane we can interpret the spherical diffraction measure as the Mellin transform of the auto-correlation distribution. We show that uniform regular model sets in commutative spaces have a pure point spherical diffraction measure. The atoms of this measure are located at the spherical automorphic spectrum of the underlying lattice, and the diffraction coefficients can be characterized abstractly in terms of the so-called shadow transform of the characteristic functions of the window. In the case of the Heisenberg group we can give explicit formulas for these diffraction coefficients in terms of Bessel and Laguerre functions.


INTRODUCTION
This article is a culmination of a series of articles begun in [7] and continued in [8] in which we extend the diffraction theory of uniform regular model sets in abelian groups to the wide setting of proper homogeneous metric spaces.In [7] we introduced regular model sets in a general locally compact second countable (lcsc) group G, and with every such model set Λ we associated a dynamical system Ω Λ over G, the hull of Λ.We then established unique ergodicity of this system and deduced that certain sampling limits over Λ −1 Λ converge to a positive-definite Radon measure η Λ on G, the auto-correlation measure of Λ.In [8] we generalized these results to (weighted) regular model sets in proper homogeneous metric spaces X .If G denotes the isometry group of X and K is one of its point stabilizers, then the auto-correlation measure of such a weighted regular model set can be seen as a positive-definite Radon measure on K \G/K .The current article is concerned with certain Fourier transforms of these measures, which we call spherical diffraction measures in analogy with the abelian case.
In the classical case, where G is abelian and K = {e} is the trival subgroup, the auto-correlation measure η Λ admits a Fourier transform η Λ , which is a Radon measure on the Pontryagin dual G of G. Due to its physical interpretation [13,17], this measure is called the diffraction measure of Λ.A particular focus is on situations where this measure is discrete, see e.g.[24,23,3,2].It is one of the cornerstones of the theory of quasi-crystallographic diffraction theory that if Λ is a uniform regular model set in a locally compact abelian group, then the diffraction measure η Λ is pure point.More precisely, if Λ arises from an abelian cut-and-project scheme (G, H, Γ) with window W ⊂ H, then it follows from work of Meyer [19,20] that where Γ ⊥ ⊂ G × H denotes the dual lattice of Γ.This amounts to an exotic Poisson summation formula of the form where m G denotes Haar measure of G and (B n ) is a suitable Følner sequence of balls in G. Our ultimate goal here is to derive similar exotic summation formulas in more general situations.
While we can establish pure point diffraction in large generality, computing the diffraction coefficients explicitly will only be possible in special situations, most notably for Heisenberg groups.One problem in generalizing Meyer's theorem beyond the abelian case is that one needs a suitable notion of Fourier transform for functions on K \G/K .In [8] we considered in some details the case of the hyperbolic plane H 2 .In particular, we explained how the auto-correlation measure of a weighted regular model set Λ in H 2 can be identified with an evenly positivedefinite distribution ξ Λ on the real line.Due to exponential volume growth of the hyperbolic plane, the distribution ξ Λ is non-tempered, and hence instead of its Fourier transform one should consider its complex Fourier transform or Mellin transform.Recall that the Mellin transform of a function ϕ ∈ C ∞ c (R) is given by By a theorem of Gelfand-Vilenkin and Krein [15, Thm.II.6.5], for every evenly positive-definite distribution ξ there exists a measure µ ξ ∈ M + (C) with supp(µ ξ ) ⊂ R ∪ iR such that ), and we call such a measure a Mellin transform of ξ.We are going to establish the following hyperbolic analogue of pure point diffraction in Theorem 6.5 below: Theorem 1.1 (Pure point diffraction, hyperbolic case).Assume that Λ is a weighted uniform regular model set in H 2 .Then its auto-correlation distribution ξ Λ has a pure point Mellin transform supported on [−1, 1] ∪ iR.
The general context in which such theorems can be established is that of spherical harmonic analysis.We will briefly explain the general formalism and then focus on the case of the Heisenberg group, in which much more precise results (in particular concerning the diffraction coefficients) can be established.
From now on let G be a lcsc group, let K < G be a compact subgroup and X = K \G.Then (G, K ) is called a Gelfand pair and X is called a commutative space if the convolution subalgebra C c (G, K ) ⊂ C c (G) of bi-K -invariant functions, the so-called Hecke algebra, is commutative.For example, the hyperbolic plane is a commutative space with G = SL 2 (R) and K = SO 2 (R).Further examples of commutative spaces include Riemannian symmetric spaces, regular trees (and more generally, Bruhat-Tits buildings) and (generalized) Heisenberg groups.
With any Gelfand pair (G, K ) one associated as spherical Fourier transform as follows.Denote by S + (G, K ) the set of positive-definite spherical functions, i.e. matrix coefficients of irreducible unitary G-representations with respect to a K -invariant vector.Such functions are bounded, and with the restriction of the weak- * -topology from L ∞ (G) the space S + (G, K ) is a locally compact space.(In the case of the hyperbolic plane it is homeomorphic ([−1, 1] ∪ iR)/{±1}.)We then define the spherical Fourier transform of (G, K ) by We say that a Radon measure η on S + (G, K ) is a spherical Fourier transform of a Radon measure η on K \G/K if for every every h ∈ span{ f * g * | f , g ∈ C c (G, K )} we have h ∈ L 1 (S + (G, K ), η) and Using a classical theorem of Godement [16] concerning the existence and uniqueness of such spherical Fourier transforms we establish: Proposition 1.2 (Existence of spherical diffraction).If Λ is a weighted FLC set in a commutative space X and ν is an invariant measure on Ω × Λ := Ω Λ \ { }, then the corresponding autocorrelation measure η Λ admits a unique spherical Fourier transform η Λ .
We refer to the Radon measure η Λ on S + (G, K ) as the spherical diffraction of Λ with respect to ν.Note that in the model set case there is a unique invariant measure on Ω × Λ , hence we can simply speak of the spherical diffraction of Λ.In the case of the hyperbolic plane this spherical diffraction is precisely the Mellin transform of the auto-correlation distribution.
Unlike the situation in the case of abelian groups, it is not true that the spherical diffraction of a regular model set in a commutative space is pure point.In fact, this property depends on the model set being uniform, a property which holds automatically for model sets in abelian groups (and, more generally, for approximate lattices in nilpotent groups [6]).
Theorem 1.3 (Pure point spherical diffraction).Let Λ be a regular model set in a commutative space X .If Λ is uniform (i.e. the underlying lattice is cocompact), then Λ has pure point spherical diffraction.
If Λ is a uniform regular model set in X = K \G, associated with a cut-and-project scheme (G, H, Γ), then we can find a countable subset C ⊂ S + (G, K ) such that In fact, the set C is simply the spherical automorphic spectrum of Γ, i.e. the set of matrix coefficients associated with those irreducible subrepresentations of the G-action on L 2 ((G×H)/Γ) which contain a K -invariant vector.In the classical case, where G is abelian and K is trivial, this set will always be a dense subset of S + (G, K ) = G, but in our more general setting, new phenomena arise.For example, if Λ is a uniform weighted regular model set in the hyperbolic plane whose underlying lattice has a strong spectral gap, then η Λ has an isolated atom at the constant function 1.
While it is easy to describe the support of the spherical diffraction measure abstractly, it is often impossible to compute it explicitly.Similarly, while we have an abstract description of the diffraction coefficients in terms of the so-called shadow transform of the characteristic function of the window, for general commutative spaces there is no hope to compute these coefficients explicitly.A notable exception is given by Gelfand pairs (G, K ), for which the group G is virtually nilpotent, hence we will focus on this case for the remainder of this introduction.
The easiest case beyond the abelian case considered by Meyer is that of the Euclidean motion group G = R n ⋊ O(n) and its maximal compact subgroup K = O(n).In this case, X = K \G is Euclidean n-space, and the corresponding spherical diffraction is the "powder diffraction" considered already more than a decade ago in [1].In this case we have , where ω κ is a certain Bessel function, and if Λ is a weighted regular model set in X which arises from an irreducible ∆ ⊂ R n × R m and window W ⊂ R m , then its diffraction is given by the formula The easiest non-virtually abelian case is that of Heisenberg motion groups, and here the diffraction formula and in particular the diffraction coefficients take already a much more involved form.To describe our results, we introduce the following notation: • For d ∈ N we abbreviate V d := C d and define Then the (2d + 1)-dimensional Heisenberg group is • The space of positive-definite spherical functions decomposes into two parts as ≥0 } consists of products of Bessel functions in complete analogy to the virtually abelian case.
virtually abelian theory and is given by matrix coefficients of (infinite-dimensional) Schrödinger representations, which can be expressed in terms of the Laguerre polynomials L k of degree k and type 0 as given by Explicitly, We now fix d 1 , d 2 ∈ N. We are going to construct a model set in N d 1 = K d 1 \G d 1 as follows: • We choose lattices ∆ < V d and Ξ < R 2 such that ∆ projects densely and injectively onto V d 1 and V d 2 , Ξ projects densely and injectively onto both coordinates and such that We then obtain a lattice For example, for d 1 = d 2 = 1 we could choose For larger d, we could take products of such lattices or arithmetic lattices associated with higher degree number fields.
• Given a j , b j ∈ R, 0 ≤ j ≤ d, we define Since Γ is countable we may choose these parameters in such a way that W := I × W o does not intersect the projection of Γ to H. We then obtain a uniform regular model set and an associated uniform regular model set Λ in the Heisenberg group With this notation understood we derive in Theorem 5.17 below the following explicit formula for the spherical diffraction of Λ: Theorem 1.4 (Polyradial diffraction in Heisenberg groups).The diffraction measure η Λ of the regular model set Λ is given by the formula where the horizontal and vertical diffraction coefficients are respectively given by Note that the horizontal part is in complete analogy with the virtual abelian case, whereas the vertical part (corresponding to infinite-dimensional representations) has no counterpart in the classical theory.The diffraction formula can be interpreted as an exotic Poisson summation formula for polyradial functions on the Heisenberg group in the following way: is a polyradial continuous function with compact support on N d 1 and B n are balls in N d 1 with respect to the Koranyi norm, then This article is organized as follows.In Section 2 we recall basic facts concerning spherical harmonic analysis.In particular, we describe in Theorem 2.21(a relative version) of the classical Godement-Plancherel theorem.An elementary proof (modulo the spherical Bochner theorem) is included in Appendix A. In Section 3 this theorem is used to define spherical diffraction measures in a rather general context.Section 4 establishes pure point spherical diffraction for uniform regular (weighted) model sets as stated in Theorem 1.3.We also give a general formula for the diffraction coefficients in terms of the so-called shadow transform in Theorem 4.8.The remainder of the article is devoted to examples.In Section 5 we explicitly compute the spherical diffraction for regular model sets in Heisenberg groups as of Theorem 1.4, using certain estimates concerning Laguerre polynomials from Appendix B. In Section 6 we explain why the spherical diffraction of regular model sets in the hyperbolic plane can be identified with the Mellin transform of the underlying auto-correlation distribution and deduce Theorem 1.1.

PRELIMINARIES ON GELFAND PAIRS
In this section we set up our notation and recall some basic results conerning Gelfand pairs.Most of the material of this subsection is fairly standard and can be found in [26,11,12,14].

Notational conventions
Throughout this article, G will always denote a unimodular lcsc group G and K < G will always denote a compact subgroup.We fix a choice of Haar measure m G on G and denote by m K the Haar probability measure on K .We also denote by K p : G → K \G, p K : G → G/K and K p K : G → K \G/K the canonical projections.Starting from Subsection 2.4 we will always assume that (G, K ) is moreover a Gelfand pair (cf.Definition 2.11).Our notation follows [8], in particular we make the following conventions: Remark 2.1 (Notations concerning function spaces).If X is a lcsc space, then we denote by C c (X ), C 0 (X ) and C b (X ) the function spaces of complex-valued compactly supported continuous functions, continuous functions vanishing at infinity and continuous bounded functions respectively.
If (X , ν) is a measure space and f , g ∈ L 2 (X , ν), then we denote by Following [8], but contrary to the convention in [7], we will choose all our inner products to be anti-linear in the second variable.Given a function f : G → C we denote by f , f and f * respectively the functions on G given by f (g) := f (g), f (g) := f (g −1 ) and f * (g) := f (g −1 ).
Remark 2.2 (Notations concerning measures).We denote by M(X ) the Banach space of complex Radon measure on X .We write M b (X ) for the subspace of finite complex measures (i.e.µ with |µ|(X ) < ∞), M + (X ) for the subset of (positive) Radon measures and M + b (X ) for the space of bounded Radon measures on X .Finally we denote by Prob(X ) ⊂ M + b (X ) the space of probability measures on X .We identify µ ∈ M(X ) with the corresponding linear functional on C c (X ) and write µ( f The group G acts on functions on G by L g f (x) := f (g −1 x) and R g f (x) := f (xg), and dually on measures.
We denote these by µ → µ ♯ (in case of measures) or f → f ♯ (in case of functions).

Remark 2.4 (Actions of convolution algebras). If
is a unitary representation of G, then we denote by the same latter the associated * -representation π : L 1 (G) → B(V ) as given by For the left-and right-regular (2.1)

Remark 2.5 (Canonical identifications). Pullback induces bijections
), and we denote their inverses by f → K f and f We use the same notation also for other classes of left-, respectively bi-K -invariant functions.The isomorphism K p * K : C c (K \G/K ) → C c (G, K ) can be used to induce a convolution structure on C c (K \G/K ).For a more explicit description of this convolution structure see Definition A.9 in [8].
Remark 2.6 (Convenient approximate identities).As pointed out in [8,Remark A.12], there exist functions ρ n ∈ C c (G) with the following properties: 1 and all of the functions are supported inside a common pre-compact identity neighbourhood.
• For every 1 ≤ p < ∞ and f ∈ L p (G) we have these convergences hold uniformly, and for f ∈ C(G) they hold uniformly on compacta, in particular pointwise.
• If we set ρ n := ρ ♯ n , then we have convergence ρ n * f → f ♯ and f * ρ n → f ♯ in the same sense.
We fix such functions once and for all and refer to ( ρ n ) and (ρ n ) as convenient approximate identities in C c (G), respectively C c (G, K ).

Functions and measures of positive type
The terminology concerning positive-definite functions varies in the literature.We will use the following: Definition 2.7.Let G be a lcsc group.
(1) A function ϕ : G → C is called positive-definite if for all λ 1 , . .., λ n ∈ C and x 1 , . .., With this terminology the following hold ([14, Sec.3.3]): Firstly, every function class of positive type has a (unique) continuous representative, which we refer to as a function of positive type.Thus, by our convention, functions of positive type are continuous.Secondly, for continuous functions being positive-definite and being of positive type is equivalent.More precisely: Lemma 2.8 (Characterizations of functions of positive type).Let ϕ ∈ C(G).Then the following are equivalent:
In this case, the pair (π, u) is unique up to isomorphism, and ϕ satisfies In the sequel we denote by P(G) ⊂ C(G) the set of continuous positive-definite functions (equivalently, functions of positive type) on G.We also denote by P(G, K ) := P(G) ∩ C(G, K ) the subset of bi-K -invariant continuous positive-definite functions.From the existence of convenient approximate identities in C c (G, K ) one deduces: ) with respect to the topology of uniform convergence on compacta.In particular, P(G, K ) ∩ C c (G, K ) span a dense subspace of C c (G, K ).
Proof.(i) For all x 1 , . .., x n ∈ G and λ 1 , . .., λ n ∈ C we have i, j (ii) The span contains all elements of the form f * g * with f , g ∈ C c (G, K ) by polarization.Choosing a convenient approximate identity for g then yields the claim.
The third characterization of Lemma 2.8 motivates the following definition: Note that if µ ∈ M(G) is of positive type in the sense of Definition 2.10, then µ ♯ ∈ M(G, K ) is of positive type relative to K (for any choice of K ).

Gelfand pairs and commutative spaces
Under our standing assumptions that G is a unimodular lcsc group and K < G is a compact subgroup, the following properties of the pair (G, K ) are equivalent (see e.g.[26, Thm.9.8.1]); here for a unitary G-representation (V , π V ) we denote by V K < V the subspace of K -invariant vectors.
(Gel1) The Hecke algebra Definition 2.11.The pair (G, K ) is called a Gelfand pair if it satisfies the equivalent properties (Gel1)-(Gel5) above.In this case, the corresponding proper homogeneous space K \G is called a commutative space.

Positive-definite spherical functions and the spherical Fourier transform
From now on (G, K ) denotes a Gelfand pair.Proposition 2.12.Let ω ∈ C(G, K ).Then the following are equivalent: (S1) The associated Radon measure m ω defined by (S2) ω is not the constant 0 function and satisfies the functional equation (2.2) and ω is a joint eigenfunction for the Hecke algebra, i.e. for every f Proof.See [11,Prop. 6.1.5 and 6.1.6]and [26,Thm. 8.2.6].Definition 2.13.A function ω ∈ C(G, K ) satisfying the equivalent conditions (S1)-(S4) above is called a (G, K )-spherical function.We denote by S (G, K ) the set of spherical functions.
If f ∈ C c (G, K ), then we define the spherical transform of f as the function We will consider the restrictions of this transform to the subsets S + (G, K ) ⊂ S b (G, K ) ⊂ S (G, K ) of positive-definite, respective bounded spherical functions.Remark 2.14 (Topologies on S b (G, K ) and S + (G, K )).The space S b (G, K ) carries a natural locally compact Hausdorff topology which can be described in several ways: (i) For every ω ∈ S b (G, K ) the functional m ω extends to a continuous linear functional on L 1 (G, K ), and this defines a bijection between S b (G, K ) and the Gelfand spectrum of L 1 (G, K ).Via this identification we obtain a locally compact topology on S b (G, K ).(ii) The same topology can be described more explicitly as the restriction of the weak- * -topology on L ∞ (G) to the subset S b (G, K ), see [11,Sec. 6.4].
In the sequel we will always equip S b (G, K ) with this locally compact topology.The subspace ) turns out to be closed, hence inherits a locally compact topology by [26, Prop.9.2.9].
Under the above identification of S b (G, K ) with the Gelfand spectrum of and this extends to L 1 (G, K ).Restricting further to S + (G, K ) we obtain the following: Definition 2.15.The spherical Fourier transform of the Gelfand pair (G, K ) is the transform Since by Lemma 2.8 any ω ∈ S + (G, K ) satisfies ω = ω * , we have the explicit formula (2.3) We record for later use the formula If G is abelian and K = {e}, then the positive-definite spherical functions are precisely the characters of G and hence the spherical Fourier transform of (G, {e}) coincides with the classical Fourier transform of G.In this case we have for all ω ∈ S + (G, {e}) the formula This generalizes as follows: Lemma 2.16 (Spherical Fourier transform and translations). (2.5) Proof.By the functional equation (S2) we have The computation for the right-translation action is similar.

Spherical representations and matrix coefficients
Definition 2.17.
If (V , π V ) an irreducible unitary G-representation, then by characterization (Gel5) of a Gelfand pair (V , π V ) is K -spherical if and only if dimV K = 1.In this case the matrix coefficient is independent of the unit vector v ∈ V K used to define it, and we refer to ω V simply as the spherical matrix coefficient of V .According to [26,Thm. 8.4.8] the assignment (V , π V ) → ω V induces a bijection between the set of unitary equivalence classes of irreducible spherical representations and the set S + (G, K ) of all positive-definite spherical functions.
By definition, W ω is the unique maximal subspace of W which is isomorphic to a direct sum of copies of (V , π V ).
Recall that if (W, π W ) is any unitary representation of G, then it induces a * -representation (denoted by the same letter) and π W (L 1 (G)) preserves irreducible subspaces of W.Moreover, the subalgebra π W (L 1 (G, K )) maps W onto the subspace W K , and hence preserves the latter.The action of the subalgebra π W (C c (G, K )) on this subspace is given by the spherical Fourier transform in the following sense; here given ω ∈ S + (G, K ) we denote W K ω := W ω ∩ W K .Lemma 2.18 (Action of the Hecke algebra on spherical representations).
Proof.We start with some easy reductions: Firstly, we may assume that u is a unit vector.Secondly, it suffices to prove the claim in the case where W = W ω ∼ = I (V , π V ).Finally, by considering the various components of u in this decomposition separately, one may assume that This finishes the proof.

The Godement-Plancherel theorem
The goal of this subsection is to explain how to extend the spherical Fourier transform to certain classes of Radon measures.Proposition 2.19.Let µ ∈ M(G) be a complex measure.Then for a complex measure µ on S + (G, K ) the following three equivalent conditions are equivalent: Proof.(God1) applied to f * f * yields (God2), (God2) implies (God3) by the polarization identity, and (God3) implies (God1) by plugging in a convenient approximate identity as in Remark 2.6 for g.Note that, by definition, the spherical Fourier transform (if it exists) only depends on the restriction µ| C c (G,K) of µ to bi-K -invariant functions.We now discuss the existence and uniqueness of spherical Fourier transforms.The following is the most general statement that we will need in the current article; for this we recall from Definition 2.10 the notion of a measure of positive type relative K .

Theorem 2.21 (Godement-Plancherel, relative version). If µ ∈ M(G, K
) is of positive type relative K , then µ has a unique spherical Fourier transform µ, which is a positive Radon measure.Moreover, µ is uniquely determined by µ.This is a slight generalization of a theorem from [16].The original version is as follows: is of positive type, then µ has a unique spherical Fourier transform µ, which is a positive Radon measure.
Proof.If µ is of positive type, then µ ♯ ∈ M(G, K ) is of positive type relative K , and we have Since the spherical Fourier transform of a measure only depends on its restriction to C c (G, K ), we have thus reduced to the relative case.
We explain how Theorem 2.21 can be deduced from the more classical spherical Bochner theorem in Appendix A. For a detailed account of Godement's original proof see [10, Chapter XV, Sec.9].As the name indicates, Corollary 2.22 implies the classical Plancherel theorem for the Gelfand pair (G, K ): Corollary 2.23 (Spherical Plancherel theorem).If ν denotes the Plancherel measure of the Gelfand pair (G, K ), then the map Proof.It is immediate from (2.7) that F extends to an isometric embedding hence the corollary follows.
Note that if H < G is a closed subgroup, then Corollary 2.22 can also be applied to the Haar measure m H of H.This yields a spherical Plancherel theorem for the homogeneous space G/H.

General setting
Throughout this section, (G, K ) denotes a Gelfand pair.From now on we reserve the letter X to denote the associated commutative space X = K \G, on which G acts by g.(K h) and we denote by K π R : G → U (L 2 (X , m X )) the corresponding unitary representation.
In [8] we have introduced the notion of a translation-bounded measure µ on X .Typical examples of such measures are given by weighted model sets, i.e. measures of the form K p * δ Λ , where Λ is a regular model set in G, δ Λ is the associated Dirac comb and K p : G → K \G is the canonical projection.
Throughout this section we will assume that µ is a translation bounded measure on X satisfying the following assumptions: (H1) The punctured hull Ω × µ := Ω µ \ { } is uniformly locally bounded, cf.Section 3.3 in [8].(H2) There exists a G-invariant probability measure on Ω × µ .Both assumptions are automatically satisfied in the case of weighted model sets.We then fix a G-invariant measure ν on Ω × µ .Everything in the sequel will depend on this choice of measure.Note however that in the case of weighted model sets the invariant measure is unique.Remark 3.1 (Notation concerning the Koopman representation).We will denote by π ν the unitary representation ), as well as the associated * -representation given by If (V , π V ) is an irreducible spherical representation with spherical matrix coefficient ω = ω V ∈ S + (G, K ), then we denote by ) ω the projection onto the corresponding isotypical component and set

The periodization map
In [8] we defined a periodization map , and our standing assumption (H1) implies that this map is actually continuous.With out current notation we have It is immediate from this explicit formula that P µ takes values in the subspace C 0 (Ω × µ ) K of K -invariant functions.The goal of this subsection is to establish the following projection formula: The reason for this terminology will become apparent in Theorem 3.10 below.
The proof of the projection formula is based on the following lemma: Proof.For g ∈ G and µ ′ ∈ Ω × µ we have We thus obtain We will apply this as follows: By Lemma 2.18 and (2.4) we have The corollary follows.
We also need to use the properties of our convenient approximate identity (ρ n ) as discussed in Remark 2.6 in the following form: Lemma 3.6.For every f ∈ C c (G, K ), the sequence (ρ n * f ) converges uniformly to f in C c (G, K ) and the sequence Proof.Let U ⊂ G be a pre-compact set such that f and all of the functions f * ρ n are supported inside U. We then have the estimate Now the first factor is bounded, since Ω µ is uniformly locally bounded.Since f * ρ n → f uniformly on U, the lemma follows.
Proof of Theorem 3.2.Let (u α ) α∈I ω be an orthonormal basis of By Lemma 3.6 we have uniform convergence

By Corollary 3.5 we have for every
Since the second factor is independent of f , the theorem follows.
The proof of Theorem 3.2 yields the formula for the diffraction coefficients, but this formula is hard to evaluate in praxis.We will later find more explicit formulas in special cases.

Auto-correlation measure and spherical diffraction
In [8] we defined the notion of an auto-correlation measure η ∈ M + (K \G/K ) associated with the invariant measure ν on Ω × µ .Our standing assumption (H1) ensures that this measure is well-defined, and it is uniquely determined by the fact that for all f ∈ C c (K \G/K ) we have In fact, to obtain a formula for η( f ) we can polarize (3.1): Remark 3.7 (Construction of the diffraction measure).The auto-correlation measure η corresponds via the isomorphism ) is of positive type relative K .By the Godement-Plancherel theorem (Theorem 2.21) it thus admits a Fourier transform, which is a positive Radon measure on S + (G, K ).We denote this Fourier transform by η, and observe that by Theorem 2.21) η and consequently η are uniquely determined by η.

Definition 3.8. The measure
In view of characterization (God2) of the Fourier transform of a measure, we have: Proposition 3.9.The spherical diffraction measure η ∈ M + (S + (G, K )) is uniquely determined by the fact that for all f ∈ C c (G, K ) we have From the projection formula (Proposition 3.2) we obtain immediately the following criterion for pure point spherical diffraction: Theorem 3.10 (Complete reducibility implies pure point spherical diffraction).Assume that (L 2 (Ω × µ , ν), π ν ) is spherically completely reducible in the sense that Then the spherical diffraction measure η is given in terms of the diffraction coefficients c ν (ω) as In particular, η is a pure point measure.
Proof.Let f ∈ C c (G, K ) and recall that this implies that This shows that the measures η and ω∈S + (G,K) c ν (ω) • δ ω coincide on all functions of the form ), and since these span a dense subspace of C c (S + (G, K )), the theorem follows.
In particular, the theorem applies if L 2 (Ω × µ , ν) is completely reducible as a unitary G-representation.We will see in the next subsection that this is the case if µ is (the Dirac comb) of a weighted uniform regular model set and ν is the unique invariant measure on its hull.For nonuniform model sets, the representation L 2 (Ω × µ , ν) will not be completely reducible.In this case irreducible subrepresentations of L 2 (Ω × µ , ν) will provide some pure point spectrum, but there will also be continuous spectrum in the diffraction measure.

Pure point spherical diffraction for weighted uniform regular model sets
The goal of this subsection is to establish that weighted uniform model sets have pure point spherical diffraction.Thus let Λ = Λ(G, H, Γ,W) be a uniform regular model set in G and let π * Λ be the associated weighted model set in K \G.Recall from [8,Lemma 3.11] that K p induces a continuous G-factor map π * : Ω Λ → Ω π * Λ , and that the unique G-invariant probability measure ν on Ω π * Λ is the push-forward under π * of the unique G-invariant probability measure ν on Ω Λ .In particular, π induces an embedding In order to show that the spherical diffraction measure η of ν is pure point, it suffices to show by Theorem 3.10 that L 2 (Ω π * Λ , ν) is completely reducible.This is established in the following proposition.Proof.We established in [7] that L 2 (Ω Λ , ν) is isomorphic to the space L 2 (Y , m Y ), where Y := Γ\(G × H) and m Y denotes the unique (G × H)-invariant probability measure on Y .Since Γ is cocompact in G × H, the (G × H)-representation L 2 (Y ) is completely reducible with finite multiplicities (see e.g.[26,Thm. 7.2.5]).Since (G, K ) is a Gelfand pair, the group G is of type I (see e.g.[9, Thm.2.2]).Consequently, every irreducible unitary representation (G × H)-representation is of the form V ⊠ W where V is an irreducible unitary G-representation, W is an irreducible unitary H-representation and V ⊠ W is isomorphic to the completed tensor product of V and W with (G × H)-action given by (g, h).(v ⊗ w) = gv ⊗ hw (see e.g.[14,Thm. 7.25]).In this situation, if (w i ) i∈I is a Hilbert space basis of W then, as G-representations, Note that I is countable, since L 2 (Y ) and hence W are separable.We deduce that, as Grepresentations, each V ⊠ W and thus also L 2 (Y ) are completely reducible with countable multiplicities.
At this point we have established Theorem 1.3.The remainder of this article is devoted to a computation of the diffraction coefficients in various cases of interests.

DIFFRACTION COEFFICIENTS OF WEIGHTED UNIFORM REGULAR MODEL SETS
Throughout this section Λ = Λ(G, H, Γ,W) denotes a uniform regular model set in G (see (see [7,Def. 2.6])) constructed from a cut-and-project scheme (G, H, Γ) (see [7,Def. 2.3]) with window W. We denote by ν the unique G-invariant probability measure on Ω K p * δ Λ and by η ∈ M + (K \G/K ) its auto-correlation measure.We have seen in the previous section that the diffraction measure η ∈ M + (S + (G, K )) is pure point.In this section we consider the problem of determining its coefficients in terms of the underlying lattice Γ < G × H and window W ⊂ H.

The shadow transform
We denote by Y the homogeneous (G ×H)-space Y := Γ\(G ×H) and by m Y the unique (G ×H)invariant probability measure on Y .We denote by Γ π R the unitary G-representation ) ω the projection onto the corresponding isotypical component and set Remark 4.1 (Extending the periodization map to measurable functions).For M ∈ {G, H, G × H} denote denote by L ∞ c (M) the space of bounded measurable functions on M which vanish outside a compact set.We then have a periodization map which extends the periodization map We are going to show: Proposition 4.2 (Existence of the shadow transform).For every ω ∈ spec (G,K) (Γ) and r ∈ L ∞ c (H) there exists an element S Γ (r Collecting the constants S Γ (r)(ω) from Proposition 4.2 we can define a linear map Definition 4.3.The map S Γ is called the shadow transform of the lattice Γ.
Corollary 4.4 (L 2 -norm of a periodization).For every f ∈ C c (G, K ) and r ∈ L ∞ c (H) we have Proof.By (2.1) we have ρ * f = π R ( f )(ρ), and hence for all g ∈ G and h ∈ H we have where we can exchange sum and integral since the sum is actually finite.
By Lemma 2.18 and (2.4) we have Now recall that our convenient approximate identify (ρ n ) has been chosen so that ρ n * f → f converges uniformly.This in turn implies that P Γ ((ρ n * f ) ⊗ r) → P Γ ( f ⊗ r) uniformly, and hence in L 2 .We deduce with Corollary 4.7 that Since the second factor is independent of f , the proposition follows.

The diffraction formula
Recall from Theorem 3.10 that the diffraction measure η of the unique G-invariant measure ν on Ω π * Λ is of the form We can now express the diffraction coefficients c(ω) in terms of the Shadow transform of the underlying lattice Γ and the characteristic function 1 W of the underlying window: Theorem 4.8 (Diffraction formula for weighted model sets).The diffaction measure is given by the formula 2 if ω ∈ spec (G,K) (Γ) and c(ω) = 0 otherwise.
In praxis, the the diffraction coefficients c(ω) = S Γ (1 W )(ω) 2 are usually much easier to determine than the shadow transform itself.For example they admit the following characterization: Proposition 4.9.The diffraction coefficients c(ω) are uniquely determined by the fact that for all f ∈ C c (G, K ), Proof.By Corollary 4.18 and Proposition 4.19 in [8] we have for all f ∈ C c (G, K ),

The shadow transform as a generalized Hecke correspondence
In addition to the standing assumptions of this section assume now that G is a totally disconnected lcsc group and K < G is a compact open subgroup.In this case the shadow transform is closely related to a more classical transform in harmonic analysis, the so-called Hecke correspondence, which we recall briefly.
Denote by p G : G × H → G and p H : G × H → H the canonical projections.We consider a lattice Γ < G × H such that For the moment we do not assume that Γ is uniform.Since Γ G is dense in G and K is open, the multiplication map Γ G × K → G is onto.We denote by g → (γ g , k g ) a fixed Borel section of this map.For simplicity let us normalize the Haar measure on G such that m G (K ) = 1.

iii) i and j induce mutually inverse isomorphisms of H-representations i
Proof.We first prove (ii).Observe first that for all (g, h) ∈ G × H, This shows that the map q : H → Γ\(G × H)/K given by q(h) := Γ(e, h)K is onto.Now assume that q(h 1 ) = q(h 2 ).Then This implies that kγ = e, hence k = γ −1 ∈ Γ K , and thus h 1 = τ(γ)h 2 ∈ Γ 0 h 2 .Conversely, if h 1 ∈ Γ 0 h 2 , then q(h 1 ) = q(h 2 ).Thus q factors through a continuous bijection i as in the proposition with inverse j.Now note that H acts on Γ\(G × H)/K from the right, since it commutes with K , and that i is H-equivariant.It follows that i is open, whence i and j are mutually inverse homeomorphisms.This proves (ii) and shows in particular that Γ 0 is of finite covolume, respectively cocompact in H if and only if Γ < G × H has the corresponding property.To show (i) it thus remains to show only that Γ 0 is discrete.However, for every compact subset W ⊂ H we have , which is finite by discreteness of Γ.This finishes the proof of (i) and provides us with a unique Hinvariant probability measure m Γ 0 \H on Γ 0 \H.Now (ii) yields an H-equivariant isomorphism i * : C c (Γ\(G × H)) K → C c (Γ 0 \H), and under this identification the unique H-invariant measures on Γ\(G × H) and Γ 0 \H must correspond, hence (iii) holds.In particular, if π denote the unitary Writing out the definitions of i * , j * and π explicitly we end up with (iv).
If we assume now that Γ is cocompact, then L 2 (Γ 0 \H) decomposes under the action of the Hecke algebra as and we note by proj ω : L 2 (Γ 0 \H) → L 2 (Γ 0 \H) ω the canonical projection.
Proof.The key observation is that since Consequently, for all h ∈ C c (G, K ) and ω ∈ S + (G, K ), and hence 1 K (ω) = 1.By definition of the shadow transform we have for every f ∈ C c (G, K ).If we choose f := 1 K , then we obtain hence S Γ (r)(ω) = proj ω ( j * (P Γ 0 r)), and since j * is equivariant under the action of the Hecke algebra, the proposition follows.
If we denote by η the diffraction measure of the unique G-invariant measure on Ω π * Γ , then we obtain: Corollary 4.12 (Spherical diffraction formula for a compact-open K ).The diffraction measure η is given by the formula

Classical examples
In the case where G and H are abelian and K = {e}, the diffraction formula in Theorem 4.8 reduces to [2, Thm.9.4], which in its essence goes back to the pioneering work of Meyer [19,20].

Let us briefly explain this reduction:
Let Λ = Λ(G, H, Γ,W) be a uniform regular model set and assume that G and H are abelian.Denote by G and H the dual groups of G and H respectively, and identify the dual group of G × H with G × H.We define the dual lattice of Γ by By assumption Γ projects injectively to G and densely to H.As in [21, p.19] one deduces: Lemma 4.13.The dual lattice Γ ⊥ projects injectively to G and densely to H.
Proof.If χ ∈ Γ ⊥ is contained in the kernel of the projection to G, i.e. χ = χ 1 ⊗χ 2 , then χ 2 is trivial on the projection of Γ to H, hence on all of H by continuity, and thus χ = 1.Moreover, since Γ projects injectively to G, the map H → (G × H)/Γ is injective, and hence the dual map Γ ⊥ → H ⊥ has dense image.
We denote by Γ ⊥ G and Γ ⊥ H the images of Γ ⊥ under the canonical projections p G : G × H → G and p H : G × H → H respectively.Using the lemma we may define the Fourier transform of f with respect to m G .We normalize the Haar measure m H on H so that Γ has covolume 1 in G × H and use the same symbol to denote the Fourier transform with respect to m H .We denote by m Y the corresponding Haar probability measure on Y := Γ\(G×H).
Every ξ ∈ Γ ⊥ defines a Γ-invariant function on G × H, hence descends to a function Γ ξ on Y , and the functions We thus obtain Note that this is not just a formal consequence of the Poisson summation formula, since 1 W is not smooth, but it is similar in spirit.In any case, we obtain from Proposition 4.9 the formula which is Meyer's formula.We now discuss an extension of this formula which appears (with different notation and under the name of "powder diffraction") in [1].For this let N = R d , H = R m and K = O(d), so that G = K ⋉ N is the isometry group of the Euclidean plane, and (G, K ) is a Gelfand pair with K \G = R n .Given a character ξ ∈ N we denote by q ξ the Bessel function and extend it to a positive-definite spherical function ω ξ : G → C on G by ω ξ (k o , n) := q ξ (n).We then obtain an identification The spherical Fourier transform of (G, K ) relates to the usual Fourier transform of N as follows.We have an isomorphism Now let Γ o < N × H be a lattice which projects injectively to N and densely to H; then the image and (G, H, Γ) is a cut-and-project scheme.We now pick a regular window W in H so that Λ = Λ(G, H, Γ,W) is a regular model set and and since f o (ξ 1 ) = f (ω ξ 1 ) we deduce from Proposition 4.9 that which can be seen as a version of the "spherical Poisson summation formula" [1] for regular model sets.

Non-classical examples
Since the formulas in the previous subsection were already known, the question arises to which other classes of examples our general diffraction formula can be applied to.In order to run our machinery we need: (1) a Gelfand pair (G, K ) with K compact; (2) another lcsc group H such that G × H admits a lattice Γ which projects injectively to G and densely to H.
For simplicity let us also assume that (3) G and H are connected Lie groups and the manifold G/K is simply-connected.
There are two main sources of examples for such quadruples (G, K , H, Γ).Let us first consider the case where G is amenable.In this case we have the following result: Proposition 4.14 (Cut-and-project schemes of amenable Lie groups).Assume that (G, K , H, Γ) satisfy ( 1)-( 3) above and that G is amenable.If G acts effectively on G/K , then the following hold: (i) G = N ⋊ L, where L is compact and contains K , and N is either abelian or 2-step nilpotent.
(ii) If π G (Γ) is contained in N, then H is either abelian or 2-step nilpotent and Γ is uniform.
The assumption that G acts effectively is not essential and can always be arranged by passing to a quotient.The crucial point is that assuming amenablity of G, the possible pairs (G, L) appearing in (i) can actually be classified; these are called nilmanifold pairs and all arise essentially from the 23 families of "maximal irreducible" nilmanifold pairs listed in [26,Sec. 13.4].
If we add the additional assumption that then the quadruples (G, K , H, Γ) satisfying ( 1)-( 4) with G amenable can actually be completely classified.Namely, H has to be a 2-step nilpotent Lie group, and these are well-known.Then Γ has to be a lattice in the nilpotent Lie group N × H, and hence arises from a rational basis of the Lie algebra of N × H by the construction described in [22, Remark after Thm.2.12].In each of these cases, one can try to compute explicitly the diffraction formula in terms of a given regular window W. We will carry this out for Heisenberg motion groups in Section 5.While many of the ideas work rather generally for quadruples satisfying ( 1)-( 4) above, some of our estimates are specific to the Heisenberg group.(For example, we use square-integrability of irreducible spherical representations in an essential way.)If we drop the condition that π G (Γ) < N then we can no longer classify the corresponding cut-and-project sets.Note that if we drop the assumption that G and H are connected, then we can no longer even guarantee that Γ is uniform.In fact, there exists a cut-and-project scheme (G, H, Γ) and a compact subgroup K < G such that G and H are compact-by-abelian (in particular amenable), (G, K ) is a Gelfand pair and Γ is non-uniform, see [7, p. 8] which is based on [4, Example 3.5] due to Bader, Caprace, Gelander and Monod.This shows that we are very far from classifying the possible cut-and-project sets in a commutative space G/K with G a general lcsc amenable group.
Among the examples of quadruples satisfying ( 1)-( 3) with non-amenable G, a central role is played by semisimple (or more generally, reductive) Gelfand pairs.Note that if G is any semisimple Lie group with finite center and K a maximal compact subgroup, then (G, K ) is a Gelfand pair, and K \G is a Riemannian symmetric space.In this case, there always exist both uniform and non-uniform cut-and-project schemes of the form (G, H, Γ) satisfying ( 1)- (3).For example, one can always take H = G or H = G C , the complexification of G.The assumption that Γ projects densely onto H implies by Margulis' arithmeticity theorem that Γ is an arithmetic group.This implies that all weighted model sets in Riemannian symmetric spaces are of arithmetic origin.The simplest examples arise for G = SL 2 (R), and we will discuss this case in Section 6.

Spherical functions for Heisenberg motion groups
Definition 5.1.The (2d + 1)-dimensional Heisenberg group N d = R ⊕ β d V d is the group with underlying set R × V d and multiplication is given by the formula Bi-invariant functions for this Gelfand pair correspond to polyradial functions on the Heisenberg group, whereas bi-invariant functions for K max d are given by the much smaller space of radial functions.Correspondingly, our polyradial diffraction formula is stronger than the corresponding radial diffraction formula.To obtain the latter from the former one basically has to expand the spherical functions of the larger Gelfand pair into those of the smaller Gelfand pair using the formulas from [25].We omit the details.

Remark 5.3 (Notation concerning function spaces). For every d ∈ N we have an isomorphism of * -algebras ι
It extends continuously to all L p -spaces for 1 ≤ p < ∞ and preserves smooth functions.
We now recall the classification of spherical functions for the Gelfand pair associated with the minimal Heisenberg motion group for a fixed d ∈ N. We need the following notion: is called the τ-twisted convolution of ϕ and ψ.
It is not hard to see that ϕ * τ ψ ∈ L 1 (V ) and it follows from The following result is significantly harder (see [25,Proposition 1.3.4]).
Proposition 5.5.For every τ = 0 we have L Note that for τ = 0 the convolution * 0 is just the usual convolution, and hence the proposition does not extend to the case τ = 0.The relation between twisted convolution and the Heisenberg group is as follows.
is the Fourier transform in the central variable evaluated at τ.We observe: Lemma 5.6.For every τ ∈ R and all f , g ∈ L 1 (N d ) we have Hence, for every τ, This shows that ( f * g) τ = f τ * τ g τ .
Applying Young's inequality we deduce in particular that Such functions can be used to parametrize the spherical functions for the Gelfand pair (G d , K d ): given by ω(k, (t, v)) := e iτt q(v). (5.3) Proof.We first prove the second statement: This establishes (5.4), and for f 1 , Since Remark 5.9 (Classification of (G d , K d )-spherical functions).We state without proof the classification of positive-definite (G d , K d )-spherical functions, which can be found e.g. in [25], see in particular Proposition 3.2.3 and the remarks after Proposition 3.2.5.It turns out that all bounded spherical function are positive-definite and that they all arise from τ-spherical functions via the construction in Proposition 5.8.Recall that the Laguerre polynomial L k of degree k and type 0 is defined by and that the zeroth Bessel function J o is defines as e ir cos θ dθ, for r ≥ 0.
For τ ∈ R \ {0}, there are countably many τ-spherical functions {q τ,α | α ∈ N d }, which can be parametrized by N d and are given by the real-valued functions For τ = 0 there are uncountably many τ-spherical function {q 0,κ | κ ∈ R d ≥0 }, which can be parametrized by R d ≥0 and are given by If we denote by ω τ,α , respectively ω 0,κ the K d -spherical functions on G d corresponding to q τ,α and q 0,κ respectively, then we have The functions q τ,α are matrix coefficients of Schrödinger representations with central character e iτ and K d acting by e i〈α,•〉 and the functions q 0,κ correspond to K d -orbits of characters as discussed in the virtually abelian case above.Note that, by the same computation as in the virtually abelian case, for

. Model sets in minimal Heisenberg motion groups
We now describe the setting that we will consider throughout the rest of this section.From now on we fix a pair of parameters d = (d 1 , d 2 ) and set Later we will also consider the group where Remark 5.10 (Cut-and-project schemes for Heisenberg motion groups).We choose lattices ∆ < V d and Ξ < R 2 such that ∆ projects densely and injectively onto V d 1 and V d 2 , Ξ projects densely and injectively onto both coordinates and such that β d (∆, ∆) ⊂ Ξ.We then obtain a lattice , and hence also a lattice in G × H, which we can further extend to a lattice in G × H. Then by construction (G, H, Γ) is a cut-and-project scheme for the minimal Heisenberg motion group G.

The Poisson summation formula and the horizontal contribution
From now on we consider functions By Lemma 5.12, in order to determine the auto-correlation measure we have to express the sum in terms of the spherical Fourier transforms of the functions ).If we formally apply the Poisson summation formula in the ξ-variables and then apply Lemma 5.6 we obtain Here the sum over ∆ is actually finite, hence the final rearrangement is legitimate.To justify the application of the Poisson summation formula, we need to ensure enough decay of the function for a fixed (δ 1 , δ 2 ) ∈ ∆.By (5.2), we know that the absolute value of this function is bounded from above by the function To justify our formal computation we thus need to ensure that this majorant is summable over Ξ ⊥ .This, however, follows from the smoothness assumptions on f 1 and f 2 , which ensure that their central Fourier transforms decay superpolynomially fast.Now recall from Lemma 4.13 that Ξ ⊥ ⊂ R 2 projects injectively onto both coordinates, hence (0, 0) is the only point in Ξ ⊥ which has a 0 coordinate.We may thus split Ξ( f 1 , f 2 ) into a sum of a horizontal part (corresponding to (τ 1 , τ 2 ) = (0, 0)) and a vertical part (5.11) The computation of the horizontal part is exactly as in the virtually abelian case: If for j ∈ {1, 2} we abbreviate g j := ( f j ) 0 , then using that * 0 is just the usual (untwisted) convolution then the Poisson summation formula (which applies in view of the regularity assumptions of the f j ) yields Now for σ j ∈ V d j we have by definition of g j and F j and (5.6) We have thus established:

Laguerre polynomials and the vertical contribution
The computation of the vertical part requires entirely new arguments based on properties of Laguerre polynomials.We have and we are going to first consider the sum S τ 1 ,τ 2 for a fixed pair (τ 1 , τ 2 ) ∈ Ξ ⊥ with τ 1 = 0 = τ 2 .We are going to use the following properties of the functions q τ,α .
(iii) For all α ∈ N d we have where the convergence is absolute in L ∞ -norms, and thus uniform in C b (V d ) K d .
The proof that the convergence is absolute in L ∞ -norm (and hence uniform) can be seen as follows.By the Cauchy-Schwartz and Bessel inequalities, the function α → 〈ϕ, q τ,α 〉〈ψ, q τ,α 〉 belongs to ℓ 1 (N d ).Since α → q τ,α L ∞ (V ) is bounded by (ii), we see that the sum of the L ∞norms of the terms in the sum above converge, hence the absolute convergence follows from the Weierstraß M-test.
We defer the proof to Appendix B, but emphasize that smoothness of f 1 and f 2 enters crucially.Using absolute convergence we may now freely reorder the sums inside S τ 1 ,τ 2 for every fixed (τ 1 , τ 2 ).In particular, if we define a function σ ∆ τ 1 ,τ 2 : then we obtain: Corollary 5.16 (Computation of the vertical contribution).

The polyradial diffraction formula
Combining the vertical and the horizontal contribution we finally obtain with Lemma 5.12 the following formula: Theorem 5.17 (Diffraction formula for the minimal Heisenberg motion group).The diffraction measure η of the regular model set where σ ∆ : Ξ ⊥ \ {(0, 0)} → C is given by (5.12).
Proof.By Lemma 5.12, Corollary 5.13 and Corollary 5.16 we have On the other hand, by Lemma 5.12 we also have and since W = I × W o we see from (5.6) and Proposition 5.8 that The theorem follows.

The auto-correlation distribution of a weighted model set in the hyperbolic plane
As explained in [8] the auto-correlation measure of a weighted model set in the hyperbolic plane can be re-interpreted as an evenly positive-definite distribution on the real line.We briefly recall the relevant results and notations.As in [8] we define elements of G := SL 2 (R) by k θ := cos2πθ sin 2πθ − sin2πθ cos 2πθ , a t := e t/2 0 0 e −t/2 and n u := 1 u 0 1 .and denote by K , A and N the respective subgroups of G consisting of these matrices.Multiplication induces a diffeomorphism A × N × K → G and thus every g ∈ G can be written uniquely as We then identify i, where the action of G on H 2 is by fractional linear transformations.The auto-correlation measure of a weighted regular model set Λ in the hyperbolic plane H 2 then gets identified with a Radon measure η on K \G/K .Now if we denote by C ∞ c (R) ev ⊂ C ∞ c (R) the subspace of even functions, then by [8, Lemma 5.2] (or [18, Theorems V.2.2 and V.2.3]) the Harish transform defines an isomorphism of * -algebras By [8, Prop.5.7] the map ) is an evenly positive-definite distribution, i.e. a continuous linear functional on C ∞ c (R) ev such that ξ(ϕ * ϕ * ) ≥ 0 for all ϕ ∈ C ∞ c (R) ev .Since it determines the auto-correlation measure of Λ, it is called the auto-correlation distribution of Λ.In view of exponential volume growth of the hyperbolic plane, this distribution is not tempered.
While tempered distribution can be studied via their Fourier transforms, for non-tempered distributions one has to consider a complex version of the Fourier transform known as the Mellin transform.Define the Paley-Wiener space PW(C) as the space of entire functions f : C → C such that for every N ∈ N there exist constants We denote by PW(C) ev ⊂ PW(C) the subspace of functions with f (−z) = f (z).Then the even Mellin transform is the isomorphism [18, Thm.V.3.4] We can thus consider an evenly positive-definite distribution as a linear functional on the even part of a Payley-Wiener space.By a classical result of Gelfand and Vilenkin (generalizing previous work of Krein) such a linear functional actually extends to a Radon measure: Theorem 6.1 (Gelfand-Vilenkin-Krein, [15, Thm.II.6.5]).If ξ : C ∞ c (R) ev → C is an evenly positive-definite distribution, then there exists a measure We refer to any measure µ ξ ∈ M + (C) which satisfies (6.2) and supp(µ ξ ) ⊂ R ∪ iR as a Mellin transform of the evenly positive-distribution ξ.For general evenly positive-definite distributions such a measure is not unique [15,Sec. II.4].Using the well-known relation between the Mellin transform, the Harish transform and the spherical Fourier transform of the Gelfand pair (G, K ) we are going to show: Theorem 6.2 (Pure point diffraction).Let Λ be a uniform regular model set in the hyperbolic plane.Then its auto-correlation distribution ξ admits a Mellin transform µ ξ which is a pure point measure.

Mellin transform vs. spherical Fourier transform
We now explain the proof of Theorem 6.2.Let η ∈ M + (G, K ) denote the auto-correlation measure of Λ.By Theorem 3.10 and Proposition 3.11 the diffraction measure η ∈ M + (S + (G, K )) is pure point.We now relate it to the auto-correlation distribution ξ of Λ. Remark 6.3 (Spherical functions).Denote by ρ : G → R the function given by ρ(a t n u k θ ) = e t/2 .Note that ρ(a t ) = e t/2 (corresponding to the half-sum of positive roots) and that ρ is right-Kinvariant.Integrating complex powers of ρ against the left-K -action provides bi-K -invariant functions and it turns out that these are precisely the spherical functions of the Gelfand pair (G, K ).Moreover, given z 1 , z 2 ∈ C we have ω z 1 = ω z 2 iff z 2 ∈ {±z 1 }.We may thus identify the spherical transform of the Gelfand pair (G, K ) with the map where the final equality follows from the fact that K gK = K g −1 K for all g ∈ G. Finally, the positive definite spherical functions are precisely those of the form ω z with z ∈ [−1, 1] ∪ iR.For z ∈ iR these correspond to spherical principal series, whereas for z ∈ [−1, 1] they correspond to spherical complementary series.We now recall the relation between the Mellin transform, the Harish transform and the spherical transform of the Gelfand pair (G, K ) [18, Thm.V.4.5]: Proof.Let z ∈ C. Using bi-K -invariance of f and right-K -invariance of ρ we obtain This proves the proposition.
If we identify S (G, K ) with C/{±1} (via ω z → {±z}), then the spherical diffraction measure η ∈ M + (S + (G, K )) corresonds to a pure point Radon measure We thus obtain the following refinement of Theorem 6.2.
Theorem 6.5.The measure µ is a Mellin transform of the auto-correlation distribution ξ.In particular, ξ has a pure point Mellin transform, which is supported on This proves that µ is a Mellin transform of ξ.

APPENDIX A. AN ELEMENTARY PROOF OF THE GODEMENT-PLANCHEREL THEOREM
This appendix is devoted to the proof of the Godement-Plancherel theorem in its most general form (Theorem 2.21).The proof is by reduction to the spherical Bochner theorem, which is easily accessible from the literature and which we recall in the next subsection.The remainder of the proof is self-contained and inspired by the proof in the abelian case as presented in the book of Berg and Forst [5].

A.1. Reminder of the spherical Bochner theorem
Recall that M + b (S + (G, K )) denotes the space of bounded (positive) Radon measures on the locally compact space S + (G, K ) of positive-definite spherical functions.Our starting point is the following spherical version of the classical Bochner theorem [26, Thm.9.3.4]:Theorem A.1 (Spherical Bochner theorem).Let ϕ ∈ P(G, K ).Then there exists a unique Definition A.2.For ϕ ∈ P(G, K ) the measure µ ϕ from Theorem A.1 is called the associated measure of ϕ.

A.2. A convenient reformulation of the Godement-Plancherel theorem
We now turn to the proof of Theorem 2.21.From now on we fix a measure µ ∈ M(G, K ) which is of positive type relative K .We have to show existence and uniqueness of a measure µ ∈ M(S + (G, K )) satisfying the equivalent conditions (God1)-(God3) from Proposition 2. 19   Using Lemma A.5 again we deduce that f (z) f (x j x −1 i z y)dm G (z)dµ(y) Let us write h j (t) = e t/2 g j (ξ j t j ).To prove (P1) it suffices to show that for all M ≥ 1, ≪ M n −M , for all M, and thus (P1) is established for g in the above product form.If g is not such a product, then one applies the same argument inductively, freezing all but one variables at a time.We now turn to the proof of (P2).By Proposition 5. Proof.For every v ∈ V we have the estimate where the last inequality holds by Cauchy-Schwarz.
(P4) The map grows at most polynomially.
The sub-multiplicative functions that we will use will be of the form If N is large enough, (P3) is clearly satisfied, and to establish (P4) we only need to show that for every r, the map grows at most polynomially (with a degree which is allowed to depend on r).Upon expanding the norm • and using the product structure of c κ , condition (B.4) amounts to proving that for every integer r, the map n → ∞ 0 t r L n (t) 2 e −t dt grows at most polynomially.We recall that

Remark 2 . 3 (
Notations concerning convolution algebras).M b (G) and L 1 (G) are Banach- *algebras under convolution.We denote by M b (G, K ) ⊂ M b (G) and L 1 (G, K ) ⊂ L 1 (G) the Banach- * -subalgebras consisting of measures and function classes which are bi-K -invariant.The spaces M(G, K ), C(G, K ), L p (G, K ) etc. are defined similarly.The * -subalgebra C c (G, K ) is called the Hecke algebra and plays a central role in the current article.Averaging over K × K defines canonical retractions M b

Corollary 4 . 5 (Lemma 4 . 6 .
Kernel of the shadow transform).A function r ∈ L ∞ c (H) is contained in the kernel of the shadow transform if and only if P Γ ( f ⊗ r) = 0 almost everywhere for all f ∈ C c (G, K ).The proof of Proposition 4.2 is in close analogy with the proof of the projection formula.Lemma 3.4 and Corollary 3.5 translate into the current setting as follows: For ρ, f ∈ C c (G, K ) and r ∈ L ∞ c (H) we have

Given d ≥ 1
we abbreviate V d := C d and denote by 〈•, •〉 and by m V d the Lebesgue measure on V d .The standard symplectic form β d on V d is given in terms of the standard Hermitian inner product 〈•, •〉 by the formula

Remark 5 . 2 (
and is clearly both left-and right-invariant.Heisenberg motion group).The group K max d := U(d) acts on N d by automorphisms via k.(t, u) = (t, ku).If K is any closed subgroup of K max d containing the diagonal subgroup K d := T d , then (K ⋉ N d , K ) is a Gelfand pair [26, Corollary 13.2.3], and K ⋉ N d is called a Heisenberg motion group.We will focus on the minimal Heisenberg motion group G d := K d ⋉N d .

∞ 0 h
j (t)L n (t)e −t dt ≪ M n −M , for all n.Since h j is compactly supported and smooth, [27, Theorem 2.1] tells us that ∞ 0 h j (t)L n (t)e −t dt ≪ M n
Definition 2.20.A measure µ satisfying the equivalent conditions of Proposition 2.19 is called a spherical Fourier transform of µ.
the dual lattice of Γ o and by (Γ ⊥ o ) N ist projection to N. From Lemma 4.13 we obtain a map ζ