On Sampling and Interpolation by Model Sets

We refine a result of Matei and Meyer on stable sampling and stable interpolation for simple model sets. Our setting is model sets in locally compact abelian groups and Fourier analysis of unbounded complex Radon measures as developed by Argabright and de Lamadrid. This leads to a refined version of the underlying model set duality between sampling and interpolation. For rather general model sets, our methods also yield an elementary proof of stable sampling and stable interpolation sufficiently far away from the critical density, which is based on the Poisson Summation Formula.


Background and plan of the article
Sampling concerns the problem of reconstructing a function f from its restriction f | Λ to a subdomain Λ. Interpolation concerns the question of how to extend a function defined on the subdomain.For particular function spaces, both problems are classical in harmonic analysis.We are interested in irregular sampling domains arising from model sets in locally compact abelian groups.
1.1.Function spaces.Consider a locally compact abelian (LCA) group G and its dual group p G. Later, we will sometimes require G to be σ-compact or metrizable.Note that σ-compactness and metrisability are dual notions [11, (24.48)], and that σ-compactness and metrisability is equivalent to second countability [6, Chap.IX, 2.9, Cor].As a consequence, second countability of G is equivalent to second countability of p G. Fix Haar measures θ G on G and θ p G on p G such that the Fourier inversion theorem [27,Thm. 4.4.5]holds.Recall that for integrable f : G Ñ C, its Fourier transform is defined by p f pχq " ş f pxqχpxq dθ G pxq for χ P p G. We also write q f pχq " ş f pxqχpxq dθ G pxq for the inverse Fourier transform.Definition 1.1 (Paley-Wiener space).Let G be an LCA group.Consider K Ă p G to be relatively compact and measurable.Then the translation invariant vector space of functions with restricted band width PW K " tf P L 2 pGq : p f | K c " 0 almost everywhereu is called the Paley-Wiener space with respect to K.
(i) We consider relatively compact K in this article only.Then G may be assumed to be metrizable.Indeed, for relatively compact K we may assume that p G is compactly generated without loss of generality, see [10,Sec. 8].
(ii) Note that f P PW K satisfies p f P L 1 p p Gq X L 2 p p Gq. Indeed, p f P L 1 p p Gq follows from Hölder's inequality by assumption on K.We thus have q p f P C 0 pGq and f " q p f in L 2 pGq by the Fourier inversion theorem.Hence any equivalence class f P PW K has a unique continuous representative.In this article, we will always consider PW K as a space of continuous functions.(iii) Note that pPW K , ¨ L 2 pGq q is a Hilbert space.Closedness is obvious from continuity of the map The above definition is also used in [14,2].Another definition is used in [18,9].If K is closed in p G, then both definitions are equivalent.This is discussed in Section 8.
The following subspace of PW K will be central in our subsequent analysis, as it provides continuous test functions for complex Radon measures which are transformable in the sense of Argabright and de Lamadrid [3,28].It is defined to be S K " PW K XL 1 pGq " tf P L 1 pGq : p f | K c " 0u .
In this article, we consider S K as a space of continuous functions.Any f P S K satisfies f P L 1 pGq X L 2 pGq and p f P C c p p Gq.Note that S K is dense in pPW K , ¨ L 2 pGq q if K is Riemann measurable, see Proposition 8.1.
1.2.Stable sampling and stable interpolation.The stable sampling condition ensures that any (continuous) Paley-Wiener function is reconstructable in a stable way from its values on a given uniformly discrete point set.We say that Λ Ă G is uniformly discrete if there exists a compact zero neighborhood B Ă G such that cardpΛ X px `Bqq ď 1 for all x P G. Definition 1.3 (stable sampling).Let G be an LCA group and let K Ă p G be relatively compact and measurable.A uniformly discrete set Λ Ă G is stable sampling for PW K if and only if there exist positive finite constants A, B such that for all f P PW K .
If Λ is stable sampling for PW K , then any f P PW K is uniquely determined by its values on Λ.Indeed, for f 1 , f 2 P PW K satisfying f 1 | Λ " f 2 | Λ we have f " f 1 ´f2 P PW K , too.Thus f L 2 pGq ď 0, which implies f " 0 pointwise by continuity of f .For a uniformly discrete set Λ, stable interpolation for PW K may be described as follows: For every square integrable sequence ϕ P 2 pΛq there exists at least one f P PW K such that f " ϕ on Λ.As discussed in Remark 1.5, this interpolation property is rephrased in the following definition.Let us denote by 1 S the indicator function of the set S. Definition 1.4 (stable interpolation).Let G be an LCA group and let K Ă p G be relatively compact and measurable.A uniformly discrete set Λ Ă G is stable interpolating for PW K if and only if there exist positive finite constants A, B such that A ϕ 2 2 pΛq ď 1 K f ϕ for all finitely supported ϕ P 2 pΛq.Here f ϕ P Cp p Gq is the discrete Fourier transform of ϕ given by f ϕ pχq " ř λPΛ ϕpλqχpλq.
Remark 1.5.As is well known, stable sampling and stable interpolating can be rephrased using Hilbert space techniques [31].Let us briefly indicate this connection in order to clarify the above definitions.For relatively compact and measurable K Ă p G consider the Hilbert space L 2 pKq, compare Lemma 8.4.The Fourier transform provides a norm preserving isomorphism between PW K and L 2 pKq.For x P G consider the "exponential" e x : p G Ñ C defined by e x pχq :" χpxq for all χ P p G. If x¨, ¨y denotes the usual scalar product in L 2 pKq, we then have for any f P PW K and any x P G that x p f , e ´x| K y " For a uniformly discrete set Λ Ă G consider EpΛq Ă L 2 pKq defined by EpΛq " te ´λ| K : λ P Λu .
The stable sampling inequalities say that EpΛq is a so-called frame in L 2 pKq.In our setting frame theory provides a reconstruction algorithm for p f P L 2 pKq, see e. g. [31, p. 159].The interpolation property is equivalent to requiring that EpΛq is a so-called Riesz-Fisher sequence.Hilbert space arguments for Riesz-Fisher sequences, see e. g. [31, p. 129], can be used to infer the following: The interpolation property is equivalent to the existence of a lower inequality in Eqn.(1.2).Moreover, existence of an upper inequality in Eqn.(1.2) is equivalent to the existence of an upper inequality in Eqn.(1.1).Is is well known that the upper inequalities hold for uniformly discrete sets Λ.We will give an elementary proof of the upper inequality for interpolation in Lemma 3.1.
1.3.Landau's necessity conditions.In Euclidean space, necessary conditions for Λ to be a set of stable sampling or of stable interpolation for PW K have been given by Landau [15].These have been extended to general LCA groups in [10].The conditions are formulated in terms of a particular uniform density.
In order to compute asymptotic frequencies, one often uses suitable averaging sequences generalizing balls, whose "boundary to bulk ratio" vanishes asymptotically.We will work with so-called van Hove sequences.For their definition, consider A Ă G and a compact set K Ă G.We denote the topological closure and interior of a set A by clpAq and intpAq, respectively.The K-boundary of A is given by for U any compact zero neighborhood, the van Hove boundary may be considered as a thickened topological boundary in that case.The value θ G pB U Aq, where θ G is the chosen Haar measure on G and A is relatively compact, can be chosen arbitrarily close to θ G pBAq for sufficiently concentrated zero neighborhoods U , compare [12,Lem. 4.4].A sequence pA n q nPN of compact sets in G of positive Haar measure is van Hove if and only if for every compact set K in G. Van Hove sequences have been introduced by Schlottmann [30] for model set analysis, where also their existence in σ-compact LCA groups is shown.In Euclidean space, any sequence of nonempty compact rectangular boxes of diverging positive inradius is a van Hove sequence, as well as any sequence of compact nonempty balls of diverging positive radius is a van Hove sequence.In general, note that the van Hove property is stable under shifting, i. e., for a van Hove sequence pA n q nPN the sequence pA n `xn q nPN is still van Hove, for any shift sequence px n q nPN .Further properties are collected in Section 9.For additional background, see e. g. [17,25,12] and references therein.
Definition 1.6 (Banach densities).Let Λ be a uniformly discrete set in a σ-compact LCA group G.
Let A " pA n q nPN be any van Hove sequence in G.We define the lower and upper Banach densities If D ÁpΛq " D ÀpΛq ": DpΛq, we say that Λ has Banach density DpΛq.
The above densities are finite due to uniform discreteness of Λ.In G " R d and with centered n-balls A n , the above densities are also called Landau-Beurling densities, compare [10,Sec. 7].In ergodic theory or number theory, the attribution to Banach is more common.Note that existence of a Banach density implies independence of the choice of the van Hove sequence.Banach densities exist for any regular model set by Fact 6.2, see Section 2 for definitions.In particular any lattice L in G, i. e., any discrete co-compact subgroup of G, has a Banach density, which equals the inverse measure of a measurable fundamental domain.For countable LCA groups and uniformly discrete Λ, it has recently been shown that the lower and upper Banach densities exist as a limit [8, Sec.2.2].
We rephrase Theorem 1" from [10] using upper and lower Banach densities.This is justified by Lemma 9.3.Fact 1.7 (Landau's necessity conditions).Let G be a σ-compact LCA group.Let K Ă p G be a relatively compact measurable set.Let Λ Ă G be a uniformly discrete set and let A be any van Hove sequence in G. Then the following hold.
1.4.Sampling and interpolation near the critical density.An interesting problem concerns the question which point sets admit stable sampling or stable interpolation arbitrarily close to the above critical density value θ p G pKq.A classic result for subsets of the line traces back to works of Ingham, Beurling and Kahane.See [18,Thm. 4.3,Thm. 4.5] for the statement and a discussion.Fact 1.8 (Stable sampling and stable interpolation on the line).Let I Ă R be a bounded interval, and let Λ Ă R be a uniformly discrete set.Let A be any van Hove sequence of intervals in R.
For so-called simple model sets in Euclidean space, a corresponding result has been proved by Matei and Meyer [18,Thm. 3.2].For general LCA groups, sampling and interpolation sets arbitrarily close to the critical density have been constructed in [1].For simple model sets in general LCA groups, an extension of [18] has been suggested by Agora et al. [2,Thm. 6].As emphasized in [14,9], these results rely on a duality between stable sampling and stable interpolation for model sets having Euclidean internal space.This duality allows to transfer the general problem to the line, where Fact 1.8 can be invoked.In Theorem 7.1, we will formulate the sampling and interpolation duality for model sets in the framework of LCA groups, where we use Fourier analysis of unbounded complex Radon measures as developed by Argabright and de Lamadrid [3].This setting is well suited, since integrable Paley-Wiener functions are test functions for transformable measures.In our proof, we follow the constructions of Matei and Meyer but argue without Bruhat-Schwartz functions, which are used in [2].A central part of the proof relies on mathematical diffraction theory as described in e. g. [24,28], see also [29] for a recent review.In comparison to previous results, our approach yields additional structural insight, which results in a refined version of the duality theorem in a more general setting.See Section 7.1 for further discussion.
As an application, we obtain the following version of the stable sampling and stable interpolation theorem for simple model sets.Cut-and-project schemes and model sets will be defined in the following section.Note that the model sets in the theorem below have a Banach density.Theorem 1.9 (Stable sampling and stable interpolation for simple model sets).Let pG, H, Lq be a complete cut-and-project scheme, where G is a σ-compact LCA group and where H " R. Let I Ă H be a bounded nonempty interval and let Λ I Ă G be the associated model set.Assume that K Ă p G is relatively compact and measurable.Then the following hold.
(i) If DpΛ I q ą θ p G pclpKqq, then Λ I is stable sampling for PW K .(ii) If DpΛ I q ă θ p G pintpKqq, then Λ I is stable interpolating for PW K .
(i) Let us compare the above result to [18,Thm. 3.2] and to [2,Thm. 6].In those references, it is assumed that K is compact, and for piiq it is assumed in addition that K is Riemann measurable.However these assumptions are not necessary, as may be seen by approximation arguments using regularity of the Haar measure.(ii) We can slightly relax the completeness assumption on the cut-and-project scheme, see Remark 7.3 and the proof of Theorem 1.9: From the four projection assumptions in the complete cut-and-project scheme pG, H, Lq described in Definition 2.1, the proof of (i) does not use that L projects densely to H.The proof of (ii) only uses that L projects densely to G and injectively to H. (iii) The works [14] and [9] construct model sets Λ I in Euclidean space which are simultaneously stable sampling and stable interpolating.This is possible if Λ I has critical Banach density θ p G pKq, and if K is Riemann measurable and a so-called bounded remainder set.In particular in [14, Lem.2.1] and [9,Thm. 3.1], the arguments leading to [18,Thm. 3.2] are reviewed.
In fact, our methods also allow to prove stable sampling and stable interpolation for general weak model sets sufficiently far away from the critical density.This will be done in the first part of the article, see Proposition 4.3 and Proposition 3.2.That part will also serve as a preparation for the second part of the article, as some of the corresponding arguments will reappear in the proof of the duality theorem.
Note added in proof: After submitting this manuscript, we became aware of recent work by Meyer [22], where he reviews his constructions for simple model sets in Euclidean space and gives an alternative proof of [18,Thm. 3.2], again by reduction to the line.Meyer's proof uses a duality principle from [21].It differs from the Duality Theorem 7.1, which is based on square integrable Paley-Wiener functions.
1.5.Outline of the article.In Section 2, we review weighted model combs and introduce our notation.Section 3 is devoted to a proof of stable interpolation for weak model sets away from the critical density.Section 4 is devoted to a proof of stable sampling for weak model sets away from the critical density.Although we discuss in Section 5 that our results are not optimal, we believe that our techniques, which presently rely on standard estimates, may be substantially refined.Section 6 discusses density formulae for model sets.This is a preparation for the proof of the duality theorem, which uses a density formula with smooth averaging functions instead of van Hove sequences.Section 7 contains a proof of the duality theorem for stable sampling and stable interpolation.We conclude that section with a proof of Theorem 1.9.The last sections collect some facts about Paley-Wiener spaces, van Hove sequences and Banach densities.1.6.Acknowledgments.The results of this article emerged from discussions during mutual visits of the authors at TU Chemnitz, FAU Erlangen-Nürnberg and TU Dortmund.The authors are grateful for support by their faculties.CS would like to thank Thomas Kalmes and Albrecht Seelmann for enlightening and fruitful discussions.CR would like to thank the participants of the conference Model sets and Aperiodic Order in Durham in September 2018, where parts of the above results were presented, for inspiring discussions.We thank the referees for very helpful comments and suggestions.

Weighted model sets
In order to set up our notation, we recall relevant notions related to model sets, following the monograph [4] where possible.
2.1.Cut-and-project schemes and weak model sets.Definition 2.1 (Cut-and-project scheme).Let G, H be LCA groups, and let L Ă G ˆH be a lattice, i. e., a discrete co-compact subgroup in G ˆH.We then call pG, H, Lq a bare cut-and-project scheme.If L projects injectively to G and densely to H, then pG, H, Lq is called a cut-and-project scheme.If L projects injectively and densely to both G and H, we call pG, H, Lq a complete cut-and-project scheme.
Remark 2.2.For a (bare) cut-and-project scheme pG, H, Lq, we will denote by π G : G ˆH Ñ G and π H : G ˆH Ñ H the canonical projections.Denseness of π H pLq can always be obtained by passing from H to the LCA group H 1 " clpπ H pLqq. Injectivity of π G | L allows to identify L and π G pLq.This identification will repeatedly be used below.The name complete cut-and-project scheme appears in [2], where also some structure theory is developed.
Model sets in G are obtained by "cutting from L all points inside some strip parallel to G" and projecting them to G.An analogous construction can be used to obtain point sets in H.This is the content of the following definition.Definition 2.3 (Weak model set).Assume that pG, H, Lq is a bare cut-and-project scheme.Take a window W Ă H and consider the projection set Λ W Ă G defined by If pG, H, Lq is a cut-and-project scheme, then .For any regular model set in a σ-compact group G, all of its pattern frequencies exist as a limit which is uniform in shifts of the chosen averaging sequence, see also below.Projection sets Λ with relatively compact window are of finite local complexity, i. e., we have Λ ´Λ Ă Λ `F , where F is some finite set.One also says that Λ is an almost lattice.
Remark 2.5 (Star map).If pG, H, Lq is a bare cut-and-project scheme and if L projects injectively to G, then the star map s : π G pLq Ñ π H pLq such that pg, spgqq P L is well defined.Hence we can write Λ W " tg P π G pLq : spgq P W u in that case.If pG, H, Lq is a complete cut-and-project scheme, then the star map is a bijection.Hence in that case we may write V Λ " th P π H pLq : s ´1phq P V u.
Denote by L 0 Ă p G ˆp H the annihilator of L, i. e., L 0 :" t 0 P p G ˆp H : 0 p q " 1 for all P Lu .
Hence we can define Lemma 2.6 (Dual cut-and-project scheme).Let pG, H, Lq be a bare cut-and-project scheme.Then p p G, p H, L 0 q is a bare cut-and-project scheme, which we call the dual cut-and-project scheme.If pG, H, Lq is a (complete) cut-and-project scheme, then p p G, p H, L 0 q is a (complete) cut-and-project scheme as well.
We write p s : π p G pL 0 q Ñ π x H pL 0 q for the star map in the dual cut-and-project scheme p p G, p H, L 0 q.In the following, given a relatively compact subset K Ă p G, we will frequently consider the projection set

2.3.
Weighted model combs and their transforms.Instead of working with point sets in G, it is sometimes advantageous to consider associated weighted Dirac combs.Let us denote by MpGq the space of complex Radon measures on G. Assume that pG, H, Lq is a bare cut-and-project scheme and that h : is a complex Radon measure on G, i. e., we have ω h P MpGq.Then ω h is called a weighted model comb, and h is called its weight function.Examples of weight functions are h P C c pHq and h " 1 W for relatively compact W Ă H, see the proof of Lemma 4.8 in [29].We have Λ W " supppω 1 W q, where supp denotes the measure-theoretic support.Another example is h P C 0 pHq of sufficiently fast decay [17].
We will also consider weighted model combs g ω " ř px,yqPL gpxqδ y for bounded g : G Ñ C such that g ω P MpHq.For relatively compact V Ă G, we have V Λ " suppp 1 V ωq.
Consider the associated dual cut-and-project scheme p p G, p H, L 0 q.As no confusion may arise, we will denote weighted model combs in that scheme by the same symbol, i. e., we write where |µ| is the total variation measure of µ, see e. g. [3,Ch. 1].Let us denote by M 8 pGq the space of translation bounded complex Radon measures on G. Observing that |ω h | " ω |h| , we conclude ω h P M 8 pGq for h P C c pHq or for h " 1 W with relatively compact W Ă H.We write L 1 pµq for the space of |µ|-integrable functions.
As weighted model combs are complex Radon measures, one might apply Fourier analysis of complex Radon measures as developed by Argabright and de Lamadrid [3,24,29,28].Weighted model sets satisfy a Poisson Summation Formula for a large class of test functions.Consider the function spaces and note S K Ă LKpGq for all measurable relatively compact K Ă p G. We have the following result.
Lemma 2.7 (Transform of weighted model combs).Let pG, H, Lq be a bare cut-and-project scheme.
Assume that h P KLpHq Y LKpHq.Then both ω h and ω q h are translation bounded complex Radon measures.Moreover the Poisson Summation Formula (2.1) ω h pgq " denspLq ¨ωq h pq gq holds for every g P KLpGq Y LKpGq.
Remark 2.8.Here denspLq is the reciprocal Haar measure of a measurable fundamental domain of L. It coincides with the density of lattice points, evaluated on any van Hove sequence.Note that denspLq ¨denspL 0 q " 1.
Proof of Lemma 2.7.Note that the Poisson Summation Formula (2.1) holds for all g P KLpGq and h P KLpHq by [28,Thm. 4.10 (ii)].In fact, the same formula also holds for g P LKpGq and h P KLpHq.Indeed, as ω h is twice transformable by [28,Thm. 4.12] and p g P KLp p Gq by the Fourier inversion theorem, we have where h : pyq " hp´yq.Here the first equation is double transformability of ω h , and the second equation is transformability of y ω h : .The statement for h P LKpHq now follows from the above arguments by double transformability of ω q h .2.4.Averages and van Hove sequences.With respect to Fourier analysis, characteristic functions of van Hove sequences behave similarly to rectangular functions of increasing width in R.
Lemma 2.9.[28,Lem,3.14]Let G be a σ-compact LCA group and let pA n q nPN be a van Hove sequence in G.We then have pointwise where 1 P p G denotes the identity character.

An interpolation upper bound.
As is well known, the interpolation upper bound holds for arbitrary uniformly discrete point sets.Here we give a short direct proof based on Dirac nets and Fourier analytic arguments.These will later reappear in the proof of the Duality Theorem 7.1.
Recall that a Dirac net in a general LCA group G is a net pv j q jPJ in G such that v j P C c pGq, v j ě 0 and ş G v j dθ G " 1 for all j.Furthermore the support of v j shrinks, i. e., for every zero neighborhood U Ă G there exists j 0 P J such for all j ě j 0 we have supppv j q Ă U .For the construction of such a net see e. g. [7,Lem. 1.6.5].If the LCA group G is metrizable, we can work with Dirac sequences pv n q nPN instead of Dirac nets.Lemma 3.1 (Upper bound for interpolation from uniformly discrete sets).Let G be any LCA group.Let Λ Ă G be any uniformly discrete point set in G. Consider any measurable relatively compact K Ă p G. Then there exists a finite constant B " BpΛ, Kq such that for all finitely supported ϕ : Λ Ñ C.
Recall that f ϕ P Cp p Gq denotes the discrete Fourier transform of ϕ, which is defined by f ϕ pχq " ř λPΛ ϕpλqχpλq.
Proof.Take any Dirac net pv j q jPJ in G of positive definite functions, compare [7,Lem. 3.4.1].Fix ε P p0, 1q and choose j " jpΛ, K, εq sufficiently large such that the following two properties are satisfied.
(a) For any λ P Λ, the set supppv j q `λ contains exactly one point of Λ.This is possible as Λ is uniformly discrete.(b) We have p v j ě ε ą 0 on K.This is possible as p v j is non-negative for every j and since p p v j q jPJ converges locally uniformly to one by [7,Lem. 3.4.5].Now take any ϕ P 2 pΛq of finite support and define f j P L 2 pGq X C c pGq by Note that by the support condition on v j we have ϕ 2 2 pΛq " f j 2 L 2 pGq " p f j L 2 p p Gq .A straightforward calculation shows p f j P L 1 p p Gq X L 2 p p Gq, where ¨fϕ .
We can thus conclude for every ϕ P 2 pΛq of finite support.Thus B " v j 2 L 2 pGq {ε 2 gives an upper bound for stable interpolation.

An interpolation lower bound.
Proposition 3.2 (Interpolation lower bound).Consider any bare cut-and-project scheme pG, H, Lq, where G is metrizable.Take any relatively compact W Ă H and consider the projection set Λ W Ă G. Then there exist a compact set K " KpW q Ă p G and a positive constant A " ApW, Kq such that for all finitely supported ϕ : Λ W Ñ C.
Remark 3.3.The following proof shows how to obtain suitable K Ă p G.Here we summarize the construction.Fix any compact unit neighborhood V Ă p G and take non-negative and positive definite Gq by Hölder's inequality and 1 K ě g K ě 0.Here we denote by f ˚g the convolution of f and g.Now fix a zero neighborhood B Ă G such that pΛ W ´ΛW q X B " t0u and choose relatively compact K Ă p G such that The positive constant A in the proposition is then given by For example, K may be taken as a member of any van Hove sequence pK n q n of sufficiently large index.
To see this, we may argue as in the proof of [28,Prop. 3.14].Note first that van Hove sequences exist in p G as p G is metrizable.Furthermore pK n zB V K n q nPN is also a van Hove sequence, which can be seen by straightforward estimates.We can then use Lemma 2.9 to infer z g Kn z g Kn p0q Ñ 1 t0u as n Ñ 8. Moreover the latter functions are dominated by the non-negative continuous function x ϕ V P LKpGq.In fact we have x ϕ V P L 1 pω 1 W ´W q, since 1 W ´W may be majorised by suitable h P KLpHq, and since we have x ϕ V P L 1 pω h q by Lemma 2.7.Hence the claim follows from dominated convergence.
Proof of Proposition 3.2.Note Λ W ´ΛW Ă Λ W ´W , where the latter set is uniformly discrete as W is relatively compact.We may thus choose any zero neighborhood B Ă G such that pΛ W ´ΛW q X B " t0u.Take any g K P KLp p Gq such that 1 K ě g K ě 0, compare Remark 3.3.We can then write The latter sum can be simplified as follows.We first note that for non-negative h : G Ñ R we have uniformly in λ P Λ W the estimate We can now estimate the above sum using an argument as in the proof of Young's inequality [11,Thm. 20.18] where p " q " 2 and r " 1 by If L projects injectively to G, then the last inequality is an equality.Collecting the above estimates, we arrive at The term in brackets will be positive if K is a member of any van Hove sequence of sufficiently large index, compare Remark 3.3. 2P LKpGq, and we have q g " f ¨f " q f ˚q f " q f ˚r q f .Here we used the notation r f pxq " f p´xq.Using the Poisson Summation Formula Lemma 2.7, we can write ÿ λPΛ W gpλq ď ω 1 W pgq ď ω h pgq " denspLq ¨ωq h pq gq " denspLq ¨ωq h p q f ˚r q f q .
This first inequality is an equality if L projects injectively to G. Now we can apply a standard estimate, using that ω q h is a complex Radon measure and that suppp q f ˚r q f q Ă K ´K.We obtain where the second estimate relies on Young's inequality.The finite constant B " BpK, W q can be chosen independently of the continuous function f P PW K by B " denspLq ¨ω| q h| pK ´Kq.

4.2.
A sampling lower bound.
Assume that H is σ-compact and that L projects both injectively and densely to G. Take any relatively compact K Ă p G. Then there exist compact W " W pKq Ă H and a positive constant for any f P PW K .
Remark 4.4.The following proof also shows how to obtain suitable windows W .Here we summarize the construction, which is analogous to that in Remark 3.3.Fix any compact zero neighborhood U Ă H and take non-negative and positive definite ϕ U P C c pHq such that supppϕ U q Ă U and Choose some compact zero neighborhood B Ă p H containing only the origin of the uniformly discrete projection set K´K Λ Ă p H. Now take compact W Ă H such that Note that H admits a Hove sequence pW n q nPN since H is σ-compact.One may choose W " W n for n sufficiently large, compare the arguments in Remark 3.3.In that case, the positive constant A can be chosen as Proof of Proposition 4.3.Define h W P KLpHq as in Remark 4.4.Take f P PW K and note g " |f | 2 P LKpGq.Using the Poisson Summation Formula Lemma 2.7, we can write In the first equation, we used that L projects injectively to G. Note that the projection set K´K Λ Ă p H is uniformly discrete as K ´K is relatively compact.Take any relatively compact unit neighborhood B Ă p H containing only the unit in K´K Λ.Noting p q f ˚r q f qp1q " f 2 2 and } h W peq " θ H pW zB U W q, we can write In the second equality, we used that L 0 projects injectively to p H, which is equivalent to π G pLq being dense in G.A standard estimate on the second term on the rhs of Eq. (4.1), which uses suppp q f ˚r q f q Ă K ´K, yields where we used Young's inequality [11,Thm. 20.18] for the last estimate.Summarizing the above estimates, we get The term in brackets will be positive if W is a member of any van Hove sequence of sufficiently large index, compare Remark 4.4.Here we use that H is σ-compact.

Lower bounds and critical density
We analyze the above method for sampling using model sets in G " R which are simple, i. e., which satisfy H " R and W Ă H an interval.We will see that the method used above does not give sharp results.
5.1.Sampling upper bound.We give an explicit upper bound for the statement in Lemma 4.1.Choose centered intervals U " r´u, us and W " r´w, ws Ă H, where u, w ą 0. Define ϕ U " 1 2u 1 U and h " 1 W ´U ˚ϕU .Then h P KLpHq is piecewise linear, and we have h " 1 on W and h " 0 on pW `U q c .The transform of h is given by We have p hp0q " 2pw `uq, and we may later use the standard estimate | p hpkq| ď p2π 2 uk 2 q ´1.Due to uniform discreteness of the projection set K´K Λ, we may assume that b ą 0 is sufficiently small such that any translation of B " r´b, bq Ă p H contains at most one point of K´K Λ.Using an interval partition of p H by shifted copies of B, we can estimate Here we used ř PN 1 p2 ´1q 2 " π 2 {8.Hence we may require the finite stable sampling upper bound to satisfy C " denspLq ¨ω| q h| p1 K´K q ď denspLq ¨ˆ2pw `uq `1 8ub 2 ˙.

5.2.
Sampling lower bound.We may proceed similarly for the upper bound case.Choose centered intervals W " r´w, ws Ă H and U " r´u, us where 0 ă u ă w.Define ϕ U " 1 2u 1 U and h W " 1 W zB U W ˚ϕU .Then h W P KLpHq is piecewise linear, and we have h W " 1 on r´pw´uq, w´us and h W " 0 on W c .We have y h W p0q " 2pw ´uq, and as above we have the standard estimate | y h W pkq| ď p2π 2 uk 2 q ´1.Due to uniform discreteness of the projection set K´K Λ, we may assume that b ą 0 is sufficiently small such that any translate of B " r´b, bq Ă p H contains at most one point of K´K Λ.Using an interval partition of p H by shifted copies of B, we can estimate as above Using the Poisson Summation Formula, we thus get A necessary condition for the rhs to be positive can be explicitly calculated as 2wb ą 1.Let us rephrase this result in terms of the density of the regular model set Λ W , which is given by denspΛ W q " denspLq ¨θH pW q, compare Fact 6.2.Assume that K is a centered interval and that K´K Λ has maximal b, i. e., p2bq ´1 " densp K´K Λq " denspL 0 q ¨θ p G pK ´Kq.We then get This is away from Landau's necessity condition denspΛ W q ě θ p G pKq by a factor of 4.

6.
Density formulae for weighted model combs The convergence is uniform in s P G.
The density formula implies that any regular model set Λ W has a uniform point density.Indeed, then h " 1 W is Riemann integrable.For a general weak model set, only upper and lower estimates can be given.The following result can be proved by approximation with regular model sets using the density formula, compare [12,Prop. 3.4].A version for G " R was already used by Meyer in [21, Rem.6.2].Fact 6.2.Let pG, H, Lq be a bare cut-and-project scheme.Assume that G is σ-compact and that L projects densely to H.For any weak model set Λ W in pG, H, Lq, any van Hove sequence A in G and any t P H, we have the estimates In particular if W is Riemann measurable, then DpΛ W `tq " denspLq ¨θH pW q.
6.2.Density formula for continuous test functions.The density formula Fact 6.1 expresses that the weighted model comb ω h has a uniform density.Indeed we have ω h pg n q Ñ denspLq ¨θH phq where g n " 1 s`An {θ G pA n q.Below we need to compute the limit in the density formula on a sequence of continuous test functions pg n q n of unbounded support.If G is σ-compact, then such sequences may be obtained using a Dirac sequence pv n q n in p G, as the sequence pp v n q n then converges locally uniformly to 1, see [7,Lem. 3.4.5].We have the following result.Proposition 6.3 (Density formula with continuous averaging functions).Let pG, H, Lq be a bare cut-and-project scheme.Assume that G is σ-compact and that L projects densely to H. Take a Dirac sequence pv n q nPN in p G and define g n P L 1 pGq by .
Then for any Riemann integrable h : H Ñ C and for every s P G we have The convergence is uniform in s P G.
The proof of Proposition 6.3 is based on a version of the Fourier-Bohr coefficient formula, compare [28, Prop.3.12], for continuous averaging functions.Proposition 6.4 (Fourier-Bohr coefficients with continuous averaging functions).Let G be a σ-compact LCA group.Take any Dirac sequence pv n q nPN in p G and define g n P L 1 pGq by .
Let µ P M 8 pGq be twice transformable and consider χ P p G. We then have for every s P G The convergence is uniform in s P G.
Proof of Proposition 6.4.We prove the proposition for χ " 1, where 1 denotes the trivial character.The general case then follows from p µptχuq " pδ χ ´1 ˚p µqpt1uq and pδ χ ´1 ˚p µq " x χµ, where ˚denotes measure convolution.Recall that p G is metrizable since G is σ-compact.Hence there exists a Dirac sequence pv n q nPN in p G. Note first that We also have | g n p1q " 1 and 0 ď | g n ď 1 by Young's inequality.The function | g n is positive definite and compactly supported, and its Fourier transform g n is integrable.As p µ P Mp p Gq is transformable, we thus have (6.2) p µ pe s ¨| g n q " q p µpδ ´s ˚gn q " µpδ ´s ˚gn q by the inversion theorem for transformable measures [3,Thm. 3.4] and by definition of the Fourier transform [3, Ch. 2].In Eqn.(6.2), recall that e s is the exponential e s pχq " χpsq.Let us consider the limit n Ñ 8 on the lhs.We first note | g n pχq Ñ 1 t1u pointwise.Indeed, this is obvious for χ " 1, and for χ ‰ 1 we choose any compact unit neighborhood K in p G such that χ R K. Then suppp| g n q Ă K for finally all n, such that ş G g n pxqχpxq dθpxq " | g n pχq " 0 for finally all n.As for finally all n.We can thus apply Lebesgue's dominated convergence theorem to obtain p µpe s ¨| g n q Ñ p µp1 t1u q " p µpt1uq as n Ñ 8 uniformly in s P G.
Proof of Proposition 6.3.Abbreviate c " denspLq ą 0, take ϕ P C c pHq and consider h " ϕ ˚r ϕ P C c pHq.Then ω h is twice transformable by [28,Thm. 4.12].We can now argue c ¨q hp1q " c ¨ωq h pt1uq " x ω h pt1uq " lim nÑ8 ω h pδ s ˚gn q uniformly in s P G.In the first equation we used that ω q h is a measure and that L 0 projects injectively to p G, which is equivalent to L projecting densely to H by Pontryagin duality.In the second equation, we used the generalized Poisson Summation Formula x ω h " c ¨ωq h , see Lemma 2.7, and for the third equation we used Proposition 6.4 as ω h P M 8 pGq is twice transformable.
The general case follows by a standard approximation argument.Consider without loss of generality any Riemann integrable h : H Ñ R and let ε ą 0. Recall from [3] that K 2 pHq " spantϕ ˚r ϕ : ϕ P C c pHqu is dense in C c pHq with respect to the supremum norm.Since h is Riemann integrable, there exist f 1 , f 2 P K 2 pHq such that f 1 ď h ď f 2 and ş pf 2 ´f1 q dθ H ď ε 2c .By the above result there exists N such that for all n ě N and i P t1, 2u we have uniformly in t P G. Thus, as ω f1 ď ω h ď ω f2 , for all n ě N we have and similarly c ż h dθ H ´ωh pδ s ˚gn q ě ´ε , both uniformly in s P G. Hence the claim of the proposition follows.
7. Sampling and interpolation duality 7.1.Statement and discussion of the duality theorem.Recall that S K " tf P L 1 pGqXL 2 pGq : The duality theorem relates, under suitable assumptions, sampling on Λ W for S K to interpolation from K Λ for PW W .We say that Λ is stable sampling for S K if Eqn. (1.1) holds for any f P S K .
Theorem 7.1 (Sampling-interpolation duality for model sets).Let pG, H, Lq be a complete cutand-project scheme.Let W Ă H and K Ă p G both be relatively compact and measurable.Then the following hold.
(i) Assume that G is σ-compact.Take any relatively compact and measurable neighborhood K Remark 7.2.In applications, it might be convenient to replace piq by the following equivalent statement.It is obtained from piq by exchanging the roles of G and H and using the dual cut-andproject scheme.
(i') Assume that H is σ-compact.Take any relatively compact and measurable neighborhood W The assumptions in the duality theorem may slightly be relaxed, as is seen by inspecting the proof of the duality theorem.(a) The proof does not use all four projection assumptions in Definition 2.1.The proof of (i) only uses that L projects injectively to G and densely to H.The proof of (i') only uses that L projects injectively to H and densely to G. The proof of (ii) does not use that L projects densely to H. (b) If K is open, we may take K 1 " K in (i).For more general K, assume that BK is nowhere dense.This comprises the cases K compact and K topologically regular, i. e., clpintpKqq " clpKq.Then, without any projection assumption, nowhere denseness implies that the set of g P p G such that pBK `gq X π p G pL 0 q " ∅ is dense G δ in p G, see the proof of [13,Prop. 8.3] or [12,Prop. 2.12].Thus by an arbitrarily small shift we may ensure BK X π p G pL 0 q " ∅.In that case, we may take K 1 " K in (i).Analogous reasoning applies to (i').(c) If W is closed, we have clpW q " W in (ii).For more general W , assume that BW is nowhere dense, which comprises the case W topologically regular.Then, without any projection assumption, by an arbitrarily small shift we may ensure BW X π H pLq " ∅.In that case Λ clpW q " Λ W in (ii).(d) Consider the case G " R or G " Z and assume that K Ă p G is a left-semiclosed multiband set, i. e., a union of finitely many pairwise disjoint intervals ra, bq.In that situation one may take K 1 " K.This is possible as the structure of K allows to approach any left interval endpoint in K from the right.Indeed, in the proof of part (i) of the Duality Theorem 7.1, one may take η n P p G -R to be η n " 1{n in Eqn.(7.3) to see that K 1 " K is a possible choice.This has been used in [14, Lem.2.1] and generalized in [9] to the multi-dimensional case.
Tracing function spaces in the proof of the Duality theorem part (i) Note that our Duality Theorem relates PW K{W and S W {K in a particular way.This is in contrast to the duality theorems in [14,9,2], which all relate PW K{W and PW W {K .In fact those duality theorems are special cases of Theorem 7.1, as can be seen using Proposition 7.4 and the previous remark.
Proposition 7.4.Let G be an LCA group.Let Λ Ă G be uniformly discrete and let K Ă p G be relatively compact, measurable and Riemann measurable.If Λ is stable sampling for S K , then Λ is stable sampling for PW K .
Proof.Note that the PW K stable sampling upper bound in Eqn.(1.1) holds for general uniformly discrete sets by [10,Lem. 2], see also Lemma 4.1 for weak model sets.It thus suffices to consider the lower bound.Assume that Λ is stable sampling for S K and that K is Riemann measurable.Fix any f P PW K .Take ε ą 0 and choose f ε P S K such that f ´fε L 2 pGq ď ε, which is possible due to Proposition 8.1.By the triangle inequality we then have Moreover, using the upper bound, we can estimate This leads to the following estimate for the lower bound: As ε ą 0 was arbitrary, the claim follows.
7.2.Proof of the Duality Theorem.We now give a proof of the Duality Theorem.Since the upper bounds for stable sampling and stable interpolation hold in general, see also Section 4.1 and Section 3.1 for model sets, we only discuss the lower bounds.
Proof of (i).We assume that there is a positive constant A S such that (7.1) for all f P S K 1 , where K 1 Ą K is any given fixed relatively compact and measurable neighborhood of K. We will show that for arbitrary ϕ P 2 p K Λq with finite support we have The strategy of the proof consists in associating to ϕ P 2 p K Λq suitable f n P S K 1 , which will be defined via their Fourier transform, compare Fig. 1.
Take any ϕ P 2 p K Λq of finite support.We consider certain L 2 p p Gq approximations to ϕ ˝p s, where p s denotes the star map p s : π p G pL 0 q Ñ π x H pL 0 q.The latter is indeed well-defined as L 0 projects injectively to p G, which follows by Pontryagin duality as L projects densely to H. Take any sequence pη n q n of characters in p G converging to the identity character, see Remark 7.3 (d).Take a Dirac sequence pv n q n in p G such that v n is positive definite.This is possible as p G is metrizable by assumption.Define p f n P L 2 p p Gq X C c p p Gq to be the finite sum v n pγ ¨χ ¨ηn q ¨ϕpp spχqq .
The latter sum is indeed finite as ϕ has finite support and as K is relatively compact.Note that the support of v n is contained in any given unit neighborhood if n is sufficiently large.Thus We argue that the functions f n asymptotically provide the claimed duality.Note first that for sufficiently large n we have by Eqn.(7.1).Let us compute that the rhs is asymptotically bounded by the rhs of Eqn.(7.2).We have the explicit expression ϕpp spχqq χpxq , where f P CpGq.Now consider f ϕ P CpHq from Definition 1.4, which satisfies ϕpp spχqq p spχqpyq .
Consider the star map s : π G pLq Ñ π H pLq, which is well-defined as L projects injectively to G. Note that f coincides on π G pLq with f ϕ ˝s.Indeed, for λ P π G pLq we have ϕpp spχqq χpλq " ϕpp spχqq p spχqpspλqq " f ϕ pspλqq .
Here we used that for any χ P π p G pL 0 q and λ P π G pLq, we have 0 :" pχ, p spχqq P L 0 and :" pλ, spλqq P L, and further 1 " 0 p q " pχ, p spχqqpλ, spλqq " χpλqp spχqpspλqq ùñ χpλq " p spχqpspλqq .By outer regularity of the Haar measure θ H on H and by [12,Lem. 3.6] there are windows W ε Ą W such that θ H pBW ε q " 0 and θ H pW ε q ´θH pW q ď ε for any ε ą 0. We can thus apply the density formula Proposition 6.3 for unbounded averaging functions to conclude from Eqn. (7.5) that lim sup

Now we can compute
In order to apply the density formula Proposition 6.3, we worked in the bare cut-and-project scheme pG, H, Lq, where we used that G is σ-compact and that L projects densely to H.This combined with Eqn.(7.4) shows stable interpolation (i).
Proof of (ii).The strategy of the proof is similar to that of (i).We will associate to any f P S K suitable ϕ n P 2 p K Λq of finite support, compare Fig. 2. Our computations will then use diffraction theory of model sets.In particular, we will use commutativity of the modified Wiener diagram, as explained in [28, Sec. 2, Rem.5.6].
Consider an arbitrary f P S K .As pG, H, Lq is a bare cut-and-project scheme and p f P C c p p Gq, we have that is a weighted model comb in p H. Fix any van Hove sequence pB n q nPN in p H. Such a sequence indeed exists as H is assumed to be metrizable.Define ϕ n : π Then ϕ n is finitely supported as p f has compact support and B n is compact.Here we use that the star map p t : π x H pL 0 q Ñ π p G pL 0 q is well-defined since L 0 projects injectively to p H, which holds by Pontryagin duality as lattice L projects densely to G. Now the density formula Fact 6.1 for Here we used that p H is σ-compact and that L 0 projects densely to p G, which follows from injectivity of π H | L by Pontryagin duality.Now the stable interpolation lower bound yields an inequality, and we show that the rhs of this inequality is asymptotically bounded from above by the expression needed for stable sampling.
Consider the weighted model comb p f pχqδ η in the bare cut-and-project scheme p p G, p H, L 0 q.By standard reasoning, see e. g. [28, Prop.5.1], its autocorrelation is γ " where we used that L 0 projects densely to p G, which follows from injectivity of π H | L by Pontryagin duality.Moreover by [28,Thm. 4.10], its diffraction measure is p γ " denspL 0 q 2 ¨|f| 2 ω, where Let us consider the normalized finite sample autocorrelations where ˚denotes convolution of measures, and where r µpgq " µpr gq denotes measure reflection.As finite point measures satisfy the convolution theorem [3, Thm.3.1], we have where multiplication of absolutely continuous measures f θ H and gθ H is understood as pf θ H q¨pgθ H q " pf gqθ H .Here f ϕn : H Ñ C is the discrete Fourier transform associated to ϕ n , i. e., the finite trigonometric sum We have argued that for every h P C c pHq we have Note that the Fourier transform is continuous on the space of positive definite complex Radon measures, see [24,Lem. 1.26] for the general case and [5,Thm. 4.16] for positive measures.Since lim nÑ8 γ n " γ vaguely, we thus also have p γ " lim nÑ8 x γ n vaguely.Taking h ε P C c pHq such that 1 clpW q ď h ε and h ε Ñ 1 clpW q pointwise as ε Ó 0, we can argue where we used dominated convergence for the last equality.We thus have shown where in the last equation we again used that π G | L is injective.We combine this with Eqn.(7.6) and the definition of stable interpolating: which finishes the proof that stable sampling is implied by stable interpolation.
Reinspecting the proof of part (ii) in the duality theorem, the following observations arise.Proof of Theorem 1.9.Note that by Eqn.(6.1) the weak model set K Λ Ă p H satisfies the density formula (7.9) denspL 0 q ¨θ p G pintpKqq ď D Áp K Λq ď D Àp K Λq ď denspL 0 q ¨θ p G pclpKqq , where A is any van Hove sequence in p H. (i) By assumption we have θ p G pclpKqq ă DpΛ I q " denspLq ¨|I|, where the latter equality is the density formula for the regular model set Λ I .Choose any Riemann integrable neighborhood K 1 of clpKq such that θ p G pK 1 q ă DpΛ I q.Such a neighborhood exists by [12,Lem. 3.6].Then by the density formula Eqn.(7.9) we have |I| ą D Àp K 1 Λq.Choose compact H -R is stable interpolating for PW I 1 by Fact 1.8 (ii).We can now invoke the Duality Theorem 7.1 (ii) to infer that Λ clpI 1 q " Λ I 1 is stable sampling for S K 1 .Since K 1 is Riemann integrable, Λ I 1 is stable sampling for PW K 1 by Proposition 7.4.In particular, Λ I 1 is stable sampling for PW K .Since I 1 Ă I, we have Λ I 1 Ă Λ I .Hence also Λ I is stable sampling for PW K .Indeed, in Definition 1.3 the lower inequality continues to hold for any superset of Λ, whereas an upper inequality is satisfied for general uniformly discrete sets.
(ii) By assumption we have θ p G pintpKqq ą DpΛ I q " denspLq ¨|I|, where the latter equality is the density formula for the regular model set Λ I .Now choose an interval I 1 Ą I such that |I 1 | ą |I| and such that still denspLq ¨|I 1 | ă θ p G pintpKqq.Then by the density formula Eqn.(7.9) we have H -R is stable sampling for PW I 1 by Fact 1.8 (i).In particular K Λ is stable sampling for S I 1 .We can now invoke the Duality Theorem 7.1 (i') to infer that Λ I is stable interpolating for PW K .Let G be an LCA group and assume that K Ă p G is Riemann measurable.Then S K is dense in pPW K , ¨ L 2 pGq q.
Proof.We will use the function space K 2 pGq " spantf ˚r f : f P C c pGqu, where r f pxq " f p´xq, compare [3].Take any f P PW K .For any given ε ą 0, we define an approximation f ε P S K as follows.Take any Dirac net pv j q j in p G.
2 " f 2 2 ¨θ p G pB U Kq. Now the result follows from Riemann measurability of K and outer regularity of the Haar measure by taking U sufficiently concentrated around unity, see [12,Lem. 4.4].Choose j such that supppv j q Ă U and that p f ´p f ˚vj 2 " f ´f ¨p v j 2 ă ε.The latter can be achieved as p v j converges locally to one.Consider now p f ε :" v j ˚pϕ ¨1KzB U K q P L 2 p p Gq .
Note that in fact p f ε P C c p p Gq, as it is a convolution of two compactly supported functions in L 2 p p Gq.
By construction we have suppp p f ε q Ă K. Furthermore, we have f ε :" q p f ε P L 1 pGq since is a product of two functions in L 2 pGq.Indeed, the convolution is in L 2 pGq by Young's inequality, as q ϕ P L 1 pGq and 1 KzB U K P L 2 pGq.Altogether we conclude f ε P S K .Now we can estimate where we used Young's inequality in the fourth line.As ε ą 0 was arbitrary, this proves the claim of the proposition.
8.2.Another definition of Paley-Wiener space.If K Ă p G is closed, then there is another definition of Paley-Wiener space.Recall that for a function f P L 2 pGq, its support is defined to be supppf q " tx P G : We consider the subspace PW 1 K of PW K defined by PW 1 K " tf P L 2 pGq : suppp p f q Ă Ku .
Proposition 8.2.Let G be an LCA group.We have A proof will be given at the end of this section after Lemma 8.4.We first collect some topological properties of the support.Lemma 8.3.Let G be an LCA group, and let K Ă G be measurable.
(i) For all f P L 2 pGq, supppf q Ă G is closed.(ii) For all f P L 2 pGq, we have f " f ¨1supppfq almost everywhere.

Proof.
Ad (i).Fix f P L 2 pGq.For all x P Gz supppf q, there is an open neighborhood U x with f ¨1U 2 " 0. U x is open, hence U x Ă Gz supppf q.Thus, Gz supppf q " Ť xPGz supppf q U x is open.Ad (ii).Let f P L 2 pGq.The measure |f | 2 ¨θG is a finite Radon measure.Indeed, the map , is a continuous linear functional, since for all g P C c pGq we have where 2 is a Radon measure.We now turn to the support of f .Each x P Gz supppf q has an open neighborhood U x such that f ¨1Ux 2 " 0. Each compact C Ă Gz supppf q admits a finite subset 2 " 0. We now use regularity to conclude f ´f ¨1supppfq Ad (iii).For closed K Ă G, the set U " GzK is open, and we have 1 K ¨1U 2 " 0. By definition of supp, we conclude U Ă Gz suppp1 K q.This is equivalent to suppp1 K q Ă K.
Ad (iv).Note that each x P Kz suppp1 K q has an open neighborhood U x with 1 Ux ¨1K 2 " 0. For compact C Ă Kz suppp1 K q, there is a finite set The Haar measure θ G is inner regular, so we have for all open K and all K with θ G pKq ă 8 If G is σ-finite, we exhaust Kz suppp1 K q with countably many sets of finite measure.
Let K Ă G be measurable.We denote by L 2 pG : Kq the subspace of functions having support within K.In slight abuse of notation, we also write The relation between these spaces is described by the following result.
In particular, if K is closed in G, then L 2 pG : Kq is closed in L 2 pGq, and the spaces in (a) coincide.
Proof.We begin with (a).
‚ clpL 2 pG : Kqq Ă L 2 pKq: Since L 2 pGq Q f Þ Ñ f ¨1GzK 2 P R is continuous, the right hand side is closed.Fix f P L 2 pG : Kq.From Lemma 8.3 (ii) and supppf q Ă K, we get f ¨1GzK 2 " f ¨1supppfq ¨1GzK 2 " 0, which establishes the claim.‚ clpL 2 pG : Kqq Ą L 2 pKq: Fix f P L 2 pKq.Using the inner regularity of the Radon measure |f | 2 ¨θG , compare the proof of Lemma 8.3 (ii), we see that for a suitable sequence of compact C n Ă K, n P N. For all n P N, we know by Lemma 8.3 (iii) that suppp1 Cn q Ă C n Ă K since C n is closed, so f ¨1Cn P L 2 pG : Kq.Now Therefore, U Ă Gz supppf q, which is equivalent to supppf q Ă suppp1 K q.
For the proof of (b), we need some more prerequisites.Note first that we may assume θ G pKq ą 0 without loss of generality.Indeed, otherwise supp 1 K " ∅.This implies L 2 pG : Kq " tf P L 2 pGq : f " 0 almost everywhereu, and the claim is trivially satisfied.
1. Let K 1 :" Ť f PL 2 pG:Kq supppf q.Then clpK 1 q " suppp1 K q: From (a) we know that K 1 Ă suppp1 K q, which implies clpK 1 q Ă suppp1 K q since suppp1 K q is closed.For the reverse inclusion, fix x P suppp1 K q and any open relatively compact neighborhood U of x.As K X U has finite measure, we can use inner regularity of the Haar measure to write Therefore, there exists a compact set C Ă K X U such that 1 C 2 ą 0. We conclude from Lemma 8.3 (iv) that θ G psuppp1 C qq ě θ G pCq ą 0, so that suppp1 Hence any open relatively compact neighborhood of x has nontrivial intersection with K 1 .This implies x P clpK 1 q, and since x P suppp1 K q was arbitrary, we have suppp1 K q Ă clpK 1 q.
2. (ii) implies that K 1 is closed: Assume x n P K 1 , n P N, and x P G such that x n Ñ x.We show x P K 1 .By definition of K 1 , for each x n P K 1 there exists f n P L 2 pG : Kq such that x n P supppf n q.In particular we have f n 2 ą 0 for all n.Hence we can define f P L 2 pGq by f :" By assumption (ii) that L 2 pG : Kq is closed, we have f P L 2 pG : Kq.Now x n P supppf q for every n, because for all n, we have f ě |fn| 2 n fn 2 , and hence for any open neighborhood U of The support of f is closed, so this implies x P supppf q.As supppf q Ă K 1 , this shows that K 1 contains x and is therefore closed.
With this preparation, the proof of (b) reduces to two lines.

Van Hove Sequences and Banach densities
We collect some properties of van Hove boundaries B K A " ppK `clpAqq X clpA c qq Y ppK clpA c qq X clpAqq that will be used in the sequel.As the following proof shows, Lemma 9.1 continues to hold in the non-abelian setting.Lemma 9.1.Let G be an LCA group, and consider arbitrary sets A, K, L Ă G.We then have (i) K `clpAq Ă intpAq Y B K A Ă pK Y t0uq `clpAq (ii) intpAqzB K A Ă tx P intpAq : ´K `x Ă intpAqu (iii) B K pA `Lq Ă pB K Aq `clpLq (iv) pL `pB K Aqq Ă pB L`K Aq Y pB L Aq Proof.We will use multiplicative notation for the group operation, which is more compact than additive notation."(i)" Assume that x P KclpAq but x R intpAq.Then x P KclpAq X intpAq c " KclpAq X clpA c q Ă B K A. The second statement is obvious."(ii)" We compute intpAqzB K A " intpAqz ppKclpAq X clpA c qq Y pKclpA c q X clpAqqq " intpAq X ppKclpAqq c Y intpAqq X ppKclpA c qq c Y intpA c qq " pintpAq X pKclpA c qq c q Y pintpAq X pKclpAqq c X pKclpA c qq c q " intpAq X pKclpA c qq c Now the claim follows: Assume k ´1x R intpAq for some k P K and some x P intpAqzB K A. Then x P kintpAq c Ă KclpA c q, which is a contradiction."(iii)" Let us first consider x P KclpALq X pintpALqq c .Then x " ka for some k P K, a P clpAq and P clpLq.We show ka P intpAq c , which results in x P pKclpAq X intpAq c qclpLq Ă pB K AqclpLq.Assume ka " x ´1 P intpAq.Then x P intpAq Ă intpAqclpLq Ă intpALq, which is a contradiction.It remains to consider the case x P KpintpALqq c X clpALq.Then x " a for some a P clpAq and some P clpLq.We show a " x ´1 P KintpAq c , which proves the claim.Indeed, assume x ´1 R KintpAq c .Then x R KpintpAq q c Ą KpintpAqclpLqq c Ą KpintpALqq c , which is a contradiction."(iv)" This is proved in [25,Lem. 2.13].
The following lemma shows that van Hove sequences may be nested and may have simple topological properties.In fact they may be supersets of any relatively compact seed.Lemma 9.2.Let G be a σ-compact LCA group with Haar measure θ G .Consider any relatively compact set A Ă G. Then there exists a van Hove sequence pA n q n such that A n " clpintpA n qq, θ G pBA n q " 0, A n Ă intpA n`1 q and A Ă A n for all n.
Proof.(i) We first prove the assertion of the lemma for A " teu.Take any van Hove sequence pC n q n .We will first use pC n q n to construct a van Hove sequence pB n q n satisfying B n " clpintpB n qq and θ G pBB n q " 0. Passing to a suitable shifted subsequence will then yield pA n q n with the claimed properties.Take any open relatively compact zero neighborhood U Ă G.For every n, take a compact set B n such that C n Ă B n Ă U C n , θ G pBB n q " 0 and clpintpB n qq " B n .This is possible by the construction given in the proof of [12,Lem. 3.6].Consider any compact K Ă G.We can now estimate B K B n " pKclpB n q X clpB c n qq Y pKclpB c n q X clpB n qq Ă pKclpU qclpC n q X clpC c n qq Y pKclpC c n q X clpU qclpC n qq Ă B KclpU q C n Y pKclpC c n q X ppclpC n q X clpU qclpC n qq Y pclpC c n q X clpU qclpC n qqq Ă B KclpU q C n Y B K C n Y B clpU q C n .
Since θ G pB n q ě θ G pC n q, the van Hove property θ G pB K B n q{θ G pB n q Ñ 0 follows from the van Hove property of pC n q n .Next, we construct pA n q n as follows.Take b P B 1 and define A 1 " B 1 b ´1.Now assume that A 1 , . . ., A k satisfying the properties claimed in the statement of the lemma have been chosen.We construct A k`1 following the proof of [26,Lem. 2.6 (e)].Take n such that θ G pB A ´1 k B n q ă θ G pintpB n qq " θ G pB n q, which is possible due to the van Hove property of pB n q n .Then intpB n qzB A ´1 k B n Ă tx P intpB n q : A k x Ă intpB n qu ": S n by Lemma 9.1 (ii) and θ G pintpB n qzB A ´1 k B n q ě θ G pintpB n qq ´θG pB A ´1 k B n q ą 0 .
In particular, we may choose s P S n and define A k`1 " B n s ´1.Then A k s Ă intpB n q implies A k Ă intpB n s ´1q " intpA k`1 q.It is obvious that pA n q n is van Hove, since it is a shifted subsequence of the van Hove sequence pB n q n .(ii) Now choose any relatively compact set A Ă G, which we assume to be nonempty without loss of generality.Take any van Hove sequence pB n q n having the properties in part (i) of the proof.Then the sequence pA n q n of compact sets defined by A n " B n clpAq satisfies A Ă A n and A n Ă intpA n`1 q for all n.Take any compact K Ă G.By Lemma 9.1 and commutativity of G, we can estimate θ G pB K A n q ď θ G ppB K B n qclpAqq " θ G pclpAqpB K B n qq ď θ G ppB clpAqK B n qq `θG ppB clpAq B n qq .
This can be used to conclude that pA n q n is van Hove.Indeed, we have θ G pB K A n q θ G pA n q ď θ G pB n q θ G pB n clpAqq ¨ˆθ G pB clpAqK B n q θ G pB n q `θG pB clpAq B n q θ G pB n q ˙Ñ 0 pn Ñ 8q .
Here we used that the first factor on the rhs is bounded by 1 uniformly in n.Indeed, by translation invariance of the Haar measure we may assume e P A without loss of generality.But then B n Ă B n clpAq, implying θ G pB n q{θ G pB n clpAqq ď 1.
Following [10], we consider a transitive relation ĺ between non-negative translation bounded measures.Let µ, ν P M 8 pGq such that µ, ν ě 0. Then µ ĺ ν iff for every ε ą 0 there exists compact K Ă G such that for all compact A Ă G we have p1 ´εqµpAq ď νpK `Aq.Let us write ν ľ µ iff µ ĺ ν.We have the following result, which allows to rephrase Theorem 1" from [10] using upper and lower Banach densities.Lemma 9.3.Let G be a σ-compact LCA group with Haar measure θ G .Let A " pA n q nPN be any van Hove sequence in G. Then for any uniformly discrete set Λ Ă G with associated counting measure δ Λ and any α ě 0 the following implications hold.As ε ą 0 was arbitrary, the claim follows.
"(ii)": Consider arbitrary ε ą 0. By assumption there exists compact K Ă G such that for all compact A Ă G we have p1 ´εqδ Λ pAq ď αθ G pK `Aq.Letting A " A n `s, we have α 1 ´ε ě δ Λ pA n `sq θ G pK `An `sq " δ Λ pA n `sq θ G pA n q ¨θG pA n q θ G pK `An q , where we used translation invariance of the Haar measure.Now the second factor on the rhs tends to unity as n Ñ 8 by the van Hove property.Indeed, we may assume 0 P K and can then use Lemma 9.1 (i).This implies α ě p1 ´εqD ÀpΛq.As ε ą 0 was arbitrary, the claim follows.

Figure 2 .
Figure 2. Tracing function spaces in proof of the Duality theorem part (ii)

8 . 1 .
Topological properties of the Paley-Wiener space 8.The test function space S K .Proposition 8.1.

Lemma 8 . 4 .
Let G be an LCA group and let K Ă G be measurable.Then the following hold.(a) We have clpL 2 pG : Kqq " L 2 pKq Ă L 2 pG : suppp1 K qq with equality if and only if θ G psuppp1 K qzKq " 0. (b) The following are equivalent: (i) δ Λ ľ αθ G ùñ D ÁpΛq ě α (ii) δ Λ ĺ αθ G ùñ D ÀpΛq ď αProof."(i)": Consider arbitrary ε ą 0. By assumption there exists compact K Ă G such that for all compact A Ă G we have p1 ´εqαθ G pAq ď δ Λ pK `Aq.Letting A " A n `s, we havep1 ´εqα ď δ Λ pK `An `sq θ G pA n `sq ď δ Λ pA n `sq θ G pA n q `δΛ pB K pA n `sqq θ G pA n q ,where we used Lemma 9.1 (i) and translation invariance of the Haar measure.Now the second term on the rhs vanishes as n Ñ 8 uniformly in s by[17, Lem.9.2.(b)].This implies p1 ´εqα ď D ÁpΛq.
a weak model set and if W has nonempty interior.(iii)Λ W is called a regular model set if Λ W is a model set and if W is Riemann measurable, i. e., θ H pBW q " 0, where θ H is a Haar measure on H.We may take a window V Ă G and consider the projection set V Λ Ă H defined by V Λ " π H pL X pV ˆHqq .

4 .
Stable sampling by model sets 4.1.A sampling upper bound.Since PW K is closed, Hilbert space techniques are available.One can show that the existence of an upper bound for stable sampling is equivalent to the existence of an upper bound for stable interpolation, see[31, p. 129].A direct proof of an upper bound is, for general uniformly discrete sets, given in [10, Lem.2].In the following, we give an alternative argument for weak model sets based on the Poisson Summation Formula.It yields an explicit constant appearing in the upper bound.This serves as a preparation for the lower bound, where the following argument will be adapted.Define the finite constant B " BpW, Kq by B " denspLq ¨ω| q h| pK ´Kq, where h P KLpHq is any function satisfying h ě 1 W .We then have [28,a 4.1 (Explicit sampling upper bound for weak model sets).Consider any bare cut-andproject scheme pG, H, Lq.Take any relatively compact W Ă H and any relatively compactK Ă p G. 2 ď B ¨ f 2 L 2 pGqfor all f P PW K .Remark 4.2.A suitable function h ě 1 W may be constructed as follows.Take any compact zero neighborhood U in H and take non-negative ϕ U P L 2 pHq such that supppϕ U q Ă U and ş H ϕ U dθ H " 1.Then h " 1 W ´U ˚ϕU satisfies h ě 1 W by construction, and we have h P KLpHq by[28, Lem.3.7].Proof of Lemma 4.1.Fix any relatively compact window W Ă H and K Ă p G. Take any h P KLpHq such that h ě 1 W , compare Remark 4.2.Consider any f P PW K .Then g " |f | H pBnq ¨řyPW Xπ H pLq |f ϕn | Stable sampling and stable interpolation for simple model sets.Theorem 7.1 can now be used in conjunction with Fact 1.8 to give a proof of Theorem 1.9.
As C c p p Gq is dense in L 2 p p Gq and K 2 p p Gq is dense in C c p p Gq and the embedding C c p p Gq Ă L 2 p p Gq is continuous, also K 2 p p Gq is dense in L 2 p p Gq. Hence we may choose ϕ P K 2 p p Gq such that ϕ ´p f 2 ă ε.Choose any compact unit neighborhood U Ă p G such that p f ¨1B U K 2 ă ε.Such a neighborhood indeed exists: By Hölder's inequality we have Proof of Proposition 8.2.As the Fourier transform is a norm isomorphism between L 2 pGq and L 2 p p Gq, see e. g. [27, Thm.4.4.13], the space PW 1 K is closed if and only if L 2 p p G : Kq is closed.Now the claim is obvious from Lemma 8.4.