Landau's necessary density conditions for LCA groups
aa r X i v : . [ m a t h . F A ] M a r LANDAU’S NECESSARY DENSITY CONDITIONSFOR LCA GROUPS
KARLHEINZ GR ¨OCHENIG, GITTA KUTYNIOK, AND KRISTIAN SEIP
Abstract.
We derive necessary conditions for sampling and interpolation ofbandlimited functions on a locally compact abelian group in line with the classicalresults of H. Landau for bandlimited functions on R d . Our conditions are phrasedas comparison principles involving a certain canonical lattice. Introduction
H. Landau’s necessary density conditions for sampling and interpolation [12] maybe viewed as a general principle resting on a basic fact of Fourier analysis: The com-plex exponentials e ikx ( k in Z ) constitute an orthogonal basis for L ([ − π, π ]). Thepresent paper extends Landau’s conditions to the setting of locally compact abelian(LCA) groups, relying in an analogous way on the basics of Fourier analysis. Thetechnicalities—in either case of an operator theoretic nature—are however quitedifferent. We will base our proofs on the comparison principle of J. Ramanathanand T. Steger. [17].We recall briefly Landau’s results, suitably adapted to our approach. Let Ω be abounded measurable set in R d and let B Ω denote the subspace of L ( R d ) consistingof those functions whose Fourier transform is supported on Ω. We say that a subsetΛ of R d is uniformly discrete if the distance between any two points exceeds somepositive number. A uniformly discrete set Λ is a set of sampling for B Ω if thereexists a constant C such that k f k ≤ C P λ ∈ Λ | f ( λ ) | for every f in B Ω , and a setof interpolation for B Ω if, for each square-summable sequence { a λ } λ ∈ Λ , there is asolution f in B Ω to the interpolation problem f ( λ ) = a λ , λ in Λ.The canonical case is when Ω is a cube of side length 2 π and Λ the integer lattice Z d . Since the complex exponentials e iλ · x ( λ in Λ) constitute an orthogonal basis for L (Ω), it is immediate by the Plancherel identity that Λ is both a set of samplingand a set of interpolation for B Ω . This result scales in a trivial way: c Z d is a set of Date : November 3, 2018.2000
Mathematics Subject Classification.
Key words and phrases.
Beurling density, sampling, interpolation, homogeneous approximationproperty, locally compact abelian group.K.G. was supported by the Marie-Curie Excellence Grant MEXT-CT-2004-517154.G.K. was supported by Preis der Justus-Liebig-University Giessen 2006 and DeutscheForschungsgemeinschaft (DFG) Heisenberg fellowship KU 1446/8-1.K.S. was supported by the Research Council of Norway grant 10323200.This research is part of the European Science Foundation Networking Programme Harmonicand Complex Analysis and Its Applications (HCAA). sampling and a set of interpolation for B Ω when Ω is a cube of side length c − π . Ifwe agree that the density of the integer lattice is 1, then we have that the densityof the lattice equals (2 π ) − d times the volume of the spectrum Ω.Landau’s work may be understood as saying that this density result takes thefollowing form for general Ω and uniformly discrete sets Λ:(S) If Λ is a set of sampling for B Ω , then Λ is everywhere at least as dense asthe lattice (2 π ) − | Ω | /d Z d . (I) If Λ is a set of interpolation for B Ω , then Λ is everywhere at least as sparseas the lattice (2 π ) − | Ω | /d Z d . Landau gave precise versions of these statements in terms of the following notionof density. For h > x a point in R d , let Q h ( x ) denote the closed cube centeredat x of side length h . Then the lower Beurling density of the uniformly discreteset Λ is defined as D − B (Λ) = lim inf h →∞ inf x ∈ R d card (Λ ∩ Q h ( x )) h d , and its upper Beurling density is D + B (Λ) = lim sup h →∞ sup x ∈ R d card (Λ ∩ Q h ( x )) h d . Landau’s result says that a set of sampling Λ for B Ω satisfies D − B (Λ) ≥ (2 π ) − d | Ω | and a set of interpolation Λ for B Ω satisfies D + B (Λ) ≤ (2 π ) − d | Ω | .Given two uniformly discrete sets Λ and Λ ′ and nonnegative numbers α and α ′ ,we write α Λ (cid:22) α ′ Λ ′ if for every positive ǫ there exists a compact subset K of R d such that for every compact subset L we have(1) (1 − ǫ ) α card (Λ ∩ L ) ≤ α ′ card (Λ ′ ∩ ( K + L )) . With this notation, we have the following equivalent way of expressing Landau’sdensity conditions:(S) If Λ is a set of sampling for B Ω , then (2 π ) − d | Ω | Z d (cid:22) Λ . (I) If Λ is a set of interpolation for B Ω , then Λ (cid:22) (2 π ) − d | Ω | Z d . The latter formulation may look less appealing than that given by Landau, butit has the advantage of presenting Landau’s density conditions as a comparisonprinciple ; we note that this version does not require the use of dilations of cubes,which in general LCA groups make no sense.We take as our starting point this reformulation of Landau’s results, and in briefour plan is as follows. We need to identify, in the general setting of LCA groups,a canonical case to be used for comparison. Then, besides comparing Λ with asuitable canonical lattice, we also need to compare spectra. We will do this byestimating a general spectrum Ω in terms of a disjoint union of small “cubes”. Anontrivial point will be to clarify what are the right “cubes” and what are the“lattices” associated with such sets. The technicalities of the comparison will in A proof of the equivalence is given in Section 7 of this paper.
ANDAU’S NECESSARY DENSITY CONDITIONS 3 fact take place at this “atomic” level, and it is here that the Ramanathan–Stegercomparison lemma will play a crucial role.While our approach leads to a best possible asymptotic result in the more generalsetting of LCA groups, we lose a subtle level of precision compared to Landau’swork, which is based on estimates for the eigenvalues of a certain concentrationoperator. For Ω a finite union of real intervals, Landau obtained sharp boundsfor the number of points from a set of sampling or of interpolation to be foundin a large interval I . In these bounds appears an additional term of order log | I | ,and—as shown in [16]—this can be seen as a manifestation of the John–Nirenbergtheorem.It is worth noting that one may encounter situations in which no obvious analogueof a lattice is available. An interesting example is that of the unit sphere in R d .In a recent paper [13], J. Marzo managed to employ Landau’s method in thissetting without any explicit comparison between uniformly discrete sets. In oursetting, the group of p -adic numbers is an example of an LCA group that fails tocontain a lattice. Our approach will be to restrict to a discrete quotient on whicha meaningful comparison with a lattice can be made.The ideas of Ramanathan and Steger have been employed by many authors. Wewould in particular like to mention the basic theory developed by R. Balan, P.Casazza, C. Heil, and Z. Landau in [1]. That paper introduces a notion of densityfor frames parameterized by discrete abelian groups, such as Gabor frames. Thepresent paper is however only loosely related to [1]; we will require a more generalnotion of density, since we will be dealing with uniformly discrete sets in generalLCA groups rather than discrete abelian groups.2. Landau’s density theorem for LCA groups
We start by recalling some basic facts about locally compact abelian (LCA)groups. For more information we refer to the books [4] and [9].Let G be a locally compact abelian group; to avoid trivialities, we assume that G is non-compact . The group multiplication will be written multiplicatively as xy ,and we will use the notation xK = { y ∈ G : y = xk, k ∈ K } and KL = { y ∈ G : y = kl, k ∈ K, l ∈ L } for x ∈ G and K , L being subsets of G . A locally compactgroup G is always equipped with a Haar measure, which in the following will bedenoted by µ G . We follow the convention that the Haar measure of a compact(sub)group is normalized to be a probability measure.Let b G be the dual group of G . We write the action of a character ω ∈ b G on x ∈ G by h ω, x i . The annihilator H ⊥ ⊆ b G of a subgroup H ⊆ G is defined as H ⊥ = { χ ∈ b G : H ⊆ ker χ } . By Pontrjagin duality, we can identify G with bb G , andwe will frequently use that b H ≃ b G/H ⊥ and ( G/H ) b ≃ H ⊥ .The Fourier transform F is defined by F f ( ω ) = b f ( ω ) = Z G f ( x ) h ω, x i dµ G ( x ) , ω ∈ b G .
KARLHEINZ GR ¨OCHENIG, GITTA KUTYNIOK, AND KRISTIAN SEIP
We assume that the Haar measures on G and b G have been chosen such that F isa unitary map from L ( G ) onto L ( b G ), in accordance with Plancherel’s theorem.If Ω ⊆ b G is a measurable set of positive measure, B Ω = { f ∈ L ( G ) : supp b f ⊆ Ω } is the space of “band-limited” functions with spectrum in Ω.A subset Λ of G is called uniformly discrete if there exists an open set U such thatthe sets λU ( λ in Λ) are pairwise disjoint. The definition of sets of sampling andinterpolation given in the introduction extends without any change to the settingof LCA groups. We are interested in such sets for the space B Ω .We will assume that the dual group b G is compactly generated. This may seema rather severe restriction and means that for instance p -adic groups are excludedfrom our consideration. However, if the spectrum is relatively compact, we mayassume without loss of generality that b G is compactly generated. For a clarificationof this point, we refer to Section 8. By the structure theory of LCA groups [9], b G is then isomorphic to R d × Z m × K for a compact group K . Consequently, G isof the form G = R d × T m × D with D a (countable) discrete group. We thenselect the uniformly discrete subset Γ = Z d × { e } × D as the canonical lattice tobe used for comparison, where e is the identity element in T m . We assume thatthe Haar measure µ b G is normalized so that µ b G ([ − π, π ] d × { e } × K ) = 1.We define the relation ‘ (cid:22) ’ for uniformly discrete subsets of G as we did in (1):Given two uniformly discrete sets Λ and Λ ′ and nonnegative numbers α and α ′ , wewrite α Λ (cid:22) α ′ Λ ′ if for every positive ǫ there exists a compact subset K of G suchthat for every compact subset L we have(2) (1 − ǫ ) α card (Λ ∩ L ) ≤ α ′ card (Λ ′ ∩ KL ) . It is immediate from the definition given by (2) that the relation ‘ (cid:22) ’ is transitive,a fact that will be used repeatedly in what follows.With this notation, we may state Landau’s necessary conditions for samplingand interpolation in the context of a general LCA group as follows.
Theorem 1.
Suppose Λ is a uniformly discrete subset of the LCA group G and Ω is a relatively compact subset of b G . (S) If Λ is a set of sampling for B Ω , then µ b G (Ω) Γ (cid:22) Λ . (I) If Λ is a set of interpolation for B Ω , then Λ (cid:22) µ b G (Ω) Γ . One may think of µ b G (Ω) as the “Nyquist density”. Indeed, the relation ‘ (cid:22) ’ givesus a way of defining densities of a uniformly discrete set: The lower uniform density of Λ is defined as D − (Λ) = sup { α : α Γ (cid:22) Λ } , and its upper uniform density is D + (Λ) = inf { α : Λ (cid:22) α Γ } , with the understanding that D + (Λ) = ∞ if the set on the right hand side is empty.We will later show that both densities are always finite, and so the infimum in thedefinition of D − (Λ) is in fact a minimum, and the supremum in the definition of ANDAU’S NECESSARY DENSITY CONDITIONS 5 D + (Λ) is a maximum. With these definitions, Theorem 1 can be reformulated inthe following classical way. Theorem 1’.
Suppose Λ is a uniformly discrete subset of the LCA group G and Ω is a relatively compact subset of b G . (S) If Λ is a set of sampling for B Ω , then D − (Λ) ≥ µ b G (Ω) . (I) If Λ is a set of interpolation for B Ω , then D + (Λ) ≤ µ b G (Ω) . We will show below (Lemma 8) that when G = R d , D − (Λ) and D + (Λ) reduce tothe usual Beurling densities. Indeed, we will see that an “intermediate” formulationof the densities, valid for any LCA group G , may be obtained by replacing thecounting measure of Γ by the Haar measure µ G . We will also show that, ingeneral, D − (Λ) ≤ D + (Λ) < ∞ . A particular consequence of this bound is that D − (Γ ) = D + (Γ ) = 1, because the transitivity of the relation ‘ (cid:22) ’ implies thateither D − (Γ ) = D + (Γ ) = 1 or D − (Γ ) = D + (Γ ) = ∞ .We will return to this discussion of uniform densities in Section 7, after the proofof Theorem 1. That proof requires some preparation, to be presented in the nextthree paragraphs. The most significant ingredients are the Fourier bases for small“cubes”, given in Section 4, and the Ramanathan–Steger comparison principle,treated in Section 5. The actual proof of Theorem 1 is given in Section 6.After a consideration of the case when b G is not compactly generated in Section 8,we close in Section 9 with some additional remarks pertaining to Theorem 1.3. Square sums of point evaluations at uniformly discrete sets
The purpose of this section is mainly to show that our a priori assumption thatΛ be a uniformly discrete set implies no loss of generality. However, one piece ofthis discussion will be needed in the proof of Theorem 1. This is Lemma 2 below,which says that uniformly discrete sets generate Carleson measures in a naturalway.We may of course remove the a priori assumption that a set of interpolation beuniformly discrete, but it is easy to see that, at any rate, a set of interpolationwill be uniformly discrete. The argument is standard. We first note that we canalways solve the interpolation problem with control of norms. This means that if Λis a set of interpolation, then there exists a constant M such that the interpolationproblem f ( λ ) = a λ can be solved with f in B Ω in such a way that k f k ≤ M X λ | a λ | for every square-summable sequence { a λ } λ ∈ Λ . This well-known fact is a conse-quence of the open mapping theorem. Now assume that for every open set U in G there are points λ and λ in G such that λ − λ is in U . Solving the problem KARLHEINZ GR ¨OCHENIG, GITTA KUTYNIOK, AND KRISTIAN SEIP f ( λ ) = 1 and f ( λ ) = 0 for every other λ in Λ, we get that k f k ≤ M and1 = | f ( λ ) − f ( λ ) | ≤ Z Ω | ˆ f ( ω ) ||h ω, λ i − h ω, λ i| dµ ˆ G ( ω ) ≤ M µ ˆ G (Ω) / sup ω ∈ Ω | − h ω, λ − λ i| , which cannot hold for arbitrary U when Ω is relatively compact.The reduction from a more general definition of sets of sampling follows the samepattern as in [19, pp. 140–141]. We will therefore be brief and only mention a fewtechnical modifications. We begin with the following result on Carleson measures. Lemma 2.
Let Λ be a uniformly discrete subset of G , and assume Ω is a relativelycompact subset of G . Then there is a positive constant C such that X λ ∈ Λ | f ( λ ) | ≤ C k f k holds for every f in B Ω .Proof. The proof is identical to that of Lemma 1 in [6]. Choose a function g in L ( G ) so that b g ( ω ) = 1 for ω ∈ Ω and such that for any (symmetric) compactneighborhood U of e , the function g ♯ ( x ) = sup u ∈ U | g ( xu ) | is also in L ( G ). (Such afunction exists by [18].) If f is in B Ω , then f = f ∗ g and f ♯ ( x ) ≤ ( | f | ∗ g ♯ )( x ) for all x in G . Consequently, k f ♯ k ≤ k f k k g ♯ k for all f in B Ω . Clearly, | f ( λ ) | ≤ f ♯ ( x )whenever x ∈ λU . Since Λ is uniformly discrete, we may choose U such that X λ ∈ Λ | f ( λ ) | = X λ ∈ Λ µ G ( U ) Z λU | f ( λ ) | dµ G ( x ) ≤ X λ ∈ Λ µ G ( U ) Z λU | f ♯ ( x ) | dµ G ( x )(3) ≤ µ G ( U ) Z G | f ♯ ( x ) | dµ G ( x ) ≤ k g ♯ k µ G ( U ) k f k . This lemma and the G -invariance of B Ω imply that an inequality of the form X λ ∈ Λ | f ( λ ) | ≤ C k f k , valid for every f in B Ω , holds if and only if Λ is a finite union of uniformly discretesets. The existence of such an inequality is sometimes explicitly required in thedefinition of a set of sampling.We may now go one step further and prove that if there are positive constants c and C such that c k f k ≤ X λ ∈ Λ | f ( λ ) | ≤ C k f k ANDAU’S NECESSARY DENSITY CONDITIONS 7 holds for every f in B Ω , then there are a uniformly discrete subset Λ ′ of Λ andpositive constants c ′ and C ′ such that c ′ k f k ≤ X λ ′ ∈ Λ ′ | f ( λ ′ ) | ≤ C ′ k f k for every f in B Ω . The key ingredient in the proof of this result is the followingcontinuity property: Suppose Λ is a uniformly discrete subset of G . Then, for every ε >
0, there exists a neighborhood U of the identity e such that if λ λ ′ is amapping from Λ to G satisfying λ ′ λ − ∈ U , then we have(4) X λ ∈ Λ | f ( λ ) − f ( λ ′ ) | ≤ ε k f k for every f in B Ω .We give the short proof of (4) and refer otherwise to Lemma 3.11 of [19]. We let g be as in the proof of Lemma 2 and obtain X λ ∈ Λ | f ( λ ) − f ( λ ′ ) | ≤ X λ ∈ Λ (cid:18)Z G | f ( y ) || g ( λy − ) − g ( λ ′ y − ) | dµ G ( y ) (cid:19) ≤ X λ ∈ Λ Z G | f ( y ) | | g ( λy − ) − g ( λ ′ y − ) | dµ G ( y ) Z G | g ( λx − ) − g ( λ ′ x − ) | dµ G ( x ) . Since the translation operator g ( x ) g ( ξx ) is continuous with respect to the L -norm, the integral to the right can be made arbitrarily small by a suitable choiceof U , which is an estimate that is uniform with respect to λ and λ ′ . In the integralto the left, we may then interchange the order of summation and integration andessentially repeat the calculation made in (3) with g in place of f . With a suitablechoice of U , the resulting estimate is (4).4. Fourier bases on small “cubes”
We will in what follows rewrite sampling and interpolation properties in terms ofthe spanning properties of the resulting functions on the Fourier transform side. ByLemma 2, if Ω is relatively compact, then Λ is a set of sampling for B Ω if and only ifthe system { e λ ( ω ) = h ω, λ i χ Ω ( ω ) : λ ∈ Λ } is a frame for L (Ω) ⊆ L ( b G ). Likewise,Λ is a set of interpolation for B Ω if and only if { e λ ( ω ) = h ω, λ i χ Ω ( ω ) : λ ∈ Λ } isa Riesz sequence in L (Ω). This means that { e λ : λ ∈ Λ } is a Riesz basis in theclosed linear span of the functions { e λ } .We may at once apply this observation to the canonical lattice Γ = Z d ×{ e }× D of Theorem 1. Indeed, writing as before b G = R d × Z m × K , we note that thecharacters labelled by Γ and restricted to Ω := [ − π, π ] d × { e } × K constitute anorthonormal basis for L (Ω ). Consequently, Γ is both a set sampling and a setof interpolation for B Ω . (See also [10].)In the classical case when G = R d , this is all we need, because we can just scaleΓ to obtain Fourier bases for arbitrarily small cubes . For general LCA groups, We recall from the introduction that the motivation for such a rescaling is that we wish toapproximate an arbitrary spectrum by a union of small “cubes”.
KARLHEINZ GR ¨OCHENIG, GITTA KUTYNIOK, AND KRISTIAN SEIP we need a different approach. It is convenient to introduce some notation in orderto state the lemma to be used in place of a simple rescaling. We will say that adiscrete subgroup Γ of G is a lattice if the quotient G/ Γ is compact. A uniformlydiscrete set Γ in G will be said to be a quasi-lattice if the following holds. Thereis a compact subgroup K of b G and a lattice Υ in K ⊥ such that Γ = { b kυ } , where υ ranges over Υ and b k ∈ G ranges over a set of representatives of G/K ⊥ in G .We may identify { b k } with b K ≃ G/K ⊥ , and consequently {h k, b k i} ( k in K ) is anorthonormal basis for L ( K, µ K ). We note that every lattice Λ is in particular aquasi-lattice; just take K = { e } and Υ = Λ. Lemma 3.
Let G be an LCA group whose dual group b G is compactly generated. Forevery open neighborhood U of the identity e in b G there exists a relatively compactsubset C of U and a quasi-lattice Γ in G with the following properties: (i) L ( C ) possesses an orthogonal basis of characters restricted to C and la-belled by Γ . (ii) There exists a discrete subset D of b G such that the translates dC, d ∈ D, form a partition of b G .Proof. Since b G is compactly generated, the structure theory implies that any neigh-borhood U ⊆ b G of e contains a compact subgroup K , such that H := b G/K ≃ R d × Z m × T ℓ × F , where F is a finite group and d, m, ℓ ≥
0. See [9, Thm. 9.6].Since the canonical projection π : b G → H is an open mapping, the image of U in H contains a neighborhood of the form C = [ − ǫ/ , ǫ/ d × { } × h − N , N (cid:17) ℓ × { e } . By construction, C is a fundamental domain for the latticeΞ = ( ǫ Z ) d × Z k × Z ℓN × F ⊆ H. Consequently, L ( C ) possesses an orthogonal basis consisting of characters re-stricted to C and labelled by Υ := Ξ ⊥ .Since Ξ is a lattice in H , Υ is a lattice in b H . We may identify Υ with a subgroupof G by Υ ⊆ b H ≃ (cid:0) b G/K (cid:1)b = K ⊥ ⊆ bb G ≃ G . Consequently, by fixing representatives b k from the cosets b K , we obtain a quasi-lattice Γ = { b kυ } in G with b k ranging over b K and υ over Υ.Next, set C = π − ( C ) and define for γ in Γ and ω in b Gψ γ ( ω ) = µ b G ( C ) − / h ω, γ i χ C ( ω ) = µ b G ( C ) − / h ω, γ i χ C ( π ( ω )) . We now prove that the functions ψ γ form an orthonormal basis for L ( C ). Weassume as usual that the Haar measure of a compact subgroup K is normalized tobe a probability measure and that the Haar measure of b G/K is normalized so thatthe Weil-Bruhat formula [18] dµ b G ( ω ) = dµ K ( k ) dµ b G/K ( π ( ω )) holds. So we obtain ANDAU’S NECESSARY DENSITY CONDITIONS 9 that µ b G ( C ) = Z b G χ C ( ω ) dµ b G ( ω ) = Z H Z K χ C ( ωk ) dµ K ( k ) dµ H ( π ( ω ))= Z H χ C ( π ( ω )) dµ H ( π ( ω )) = µ H ( C )and that k ψ γ k = 1 for every γ in Γ. If γ = b kυ and γ ′ = b k ′ υ ′ are in Γ, then usingthe Weil-Bruhat formula once more, we obtain that Z b G ψ γ ( ω ) ψ γ ′ ( ω ) dµ b G ( ω )= µ b G ( C ) − Z H (cid:16) Z K h ωk, b kυ b k ′− ( υ ′ ) − i χ C ( ωk ) dµ K ( k ) (cid:17) dµ H ( π ( ω ))= δ b k, b k ′ µ b G ( C ) − Z H h π ( ω ) , υ ( υ ′ ) − i χ C ( π ( ω )) dµ H ( π ( ω )) = δ γ,γ ′ . Here we have used that h ωk, υ ( υ ′ ) − i is independent of k in K , that {h k, b k i} is anorthonormal basis for L ( K ), and that {h π ( ω ) , υ i} υ ∈ Υ is an orthogonal basis for L ( C ).Next we show that the linear span of ψ γ ( γ in Γ) is dense in L ( C ). So assumethat for some f in L ( C ) and all γ in Γ we have0 = Z b G f ( ω ) ψ γ ( ω ) dµ b G ( ω )= µ b G ( C ) − / Z H (cid:16) Z K f ( ωk ) h ωk, b k i dµ K ( k ) (cid:17) h π ( ω ) , υ i χ C ( π ( ω )) dµ H ( π ( ω )) . Since {h π ( ω ) , υ i} is an orthogonal basis for L ( C ), we find that Z K f ( ωk ) h ωk, b k i dµ K ( k ) = 0for almost all π ( ω ) in b G/K and all b k in ˆ K . We infer that f ( ωk ) = 0 for almostall ω in C and k in K , since {h k, b k i} is an orthonormal basis for L ( K ). Thus thefunctions ψ γ form an orthonormal basis for L ( C ).To show (ii) we choose a pre-image D of Ξ in b G , i.e., for each λ in Ξ, D ∩ π − ( λ )contains exactly one element. Then π ( D ) = Ξ. If dC ∩ d ′ C = ∅ for d = d ′ ( d, d ′ in D ), then π ( d ) π ( C ) ∩ π ( d ′ ) π ( C ) = λC ∩ λ ′ C = ∅ for λ = λ ′ ( λ, λ ′ in Ξ). Since C is a fundamental domain for the lattice Ξ, we conclude that λ = λ ′ . By choiceof D we also have d = d ′ , a contradiction. Thus the translates dC ( d in D ) form apartition of b G , and (ii) is proved.5. The Ramanathan–Steger comparison principle
The following lemma is a variation of an argument invented by Ramanathan andSteger [17]. Their decisive idea has been investigated quite intensively in recentyears. See [1, 3, 6, 8, 11] for a sample of references and [7] for an excellent survey.
We follow the early paper [6]. In what follows, H is a separable Hilbert space withinner product h· , ·i and norm k · k . Lemma 4.
Let Γ and Λ be uniformly discrete subsets of G . Suppose that thesequence { g γ : γ ∈ Γ } is a Riesz sequence in H and that there exists a sequence { h λ : λ ∈ Λ } so that, for fixed ǫ > and a compact set K ⊆ G , (5) dist H (cid:16) g γ , span { h λ : λ ∈ Λ ∩ γK } (cid:17) < ǫ for every γ ∈ Γ . Then for every compact set L ⊆ G we have (6) (1 − cǫ ) card (Γ ∩ L ) ≤ card (Λ ∩ LK ) . The constant c > depends only on { g γ } . In particular, c = 1 if the g γ constitutean orthonormal set.Proof. Fix a compact set L ⊆ G and set H = span { g γ : γ ∈ Γ } . Then { g γ : γ ∈ Γ } is a Riesz basis for H with dual basis { ˜ g γ : γ ∈ Γ } ⊆ H , say.Since { ˜ g γ } is also a Riesz basis, it is bounded, and so(7) c = sup γ ∈ Γ k ˜ g γ k < ∞ . If { g γ : γ ∈ Γ } is an orthonormal basis, then ˜ g γ = g γ and c = 1.Let W r ( L ) = span { g γ : γ ∈ Γ ∩ L } and W f ( KL ) = span { h λ : λ ∈ Λ ∩ KL } .Let P W r denote the orthogonal projection onto W r ( L ) and Q W f the orthogonalprojection onto W f ( LK ).Using these projections, we can recast assumption (5) as k ( I − Q W f ) g γ k < ǫ provided that γ ∈ Γ ∩ L (because in this case Λ ∩ γK ⊆ Λ ∩ KL ). Consequently,we also have(8) k ( I − P W r Q W f ) g γ k = k P W r ( I − Q W f ) P W r g γ k < ǫ for all γ ∈ Γ ∩ L .
The proof is done by estimating the trace of T = P W r Q W f P W r : H → H in twodifferent ways. First, since all eigenvalues ν k of T satisfy 0 ≤ ν k ≤
1, we have(9) tr ( T ) ≤ rank T ≤ dim (cid:16) W f ( LK ) (cid:17) ≤ card (Λ ∩ LK ) . On the other hand, using (7) and (8), we find thattr ( T ) = X γ ∈ Γ ∩ L h T g γ , ˜ g γ i = X γ ∈ Γ ∩ L (cid:16) h g γ , ˜ g γ i − h ( I − T ) g γ , ˜ g γ i (cid:17) ≥ X γ ∈ Γ ∩ L − X γ ∈ Γ ∩ L cǫ = (1 − cǫ ) card (Γ ∩ L ) . (10)The claim (6) now follows from (9) and the above. ANDAU’S NECESSARY DENSITY CONDITIONS 11
In the proof of our main theorem, we will use an orthonormal basis with theproperty that N functions are associated to each point γ in Γ. In this case we haveto count each γ in the final estimate (10) with multiplicity N . This modificationyields the following statement. Lemma 5.
Let Γ and Λ be uniformly discrete subsets of G . Suppose that thesequence { g γ,j : γ ∈ Γ , j = 1 , . . . , N } is a Riesz sequence in H and that there existsa sequence { h λ : λ ∈ Λ } so that, for fixed ǫ > and a compact set K ⊆ G , dist H (cid:16) g γ,j , span { h λ : λ ∈ Λ ∩ γK } (cid:17) < ǫ for every γ ∈ Γ and j = 1 , . . . , N . Then for every compact set L ⊆ G we have (1 − cǫ ) N card (Γ ∩ L ) ≤ card (Λ ∩ LK ) . The constant c > depends only on { g γ,j } , and c = 1 if { g γ,j } is an orthonormalset. Our application of the Ramanathan–Steger comparison lemma will require anestimate usually called the homogeneous approximation property . To state it, weintroduce the following notation. Let M x be the modulation operator defined by M x f ( ω ) = h ω, x i f ( ω ) for f ∈ L ( b G ), x ∈ G, ω ∈ b G . Lemma 6.
Let b G be compactly generated, and assume that { e λ = M λ g : λ ∈ Λ } ,with g in L ∞ (Ω) , is a frame for L (Ω) with dual frame { h λ : λ ∈ Λ } . Then forevery f in L (Ω) and ǫ > there is a compact set K ⊆ G (depending on f and ǫ )such that (11) dist H (cid:16) M x f, span { h ν : ν ∈ Λ ∩ xK } (cid:17) < ǫ for every x ∈ G .Proof. The proof is identical to the proof of Lemma 2 in [6]. Using the frameexpansion of f ∈ L (Ω), we write M x f = X λ ∈ Λ h M x f, M λ g i h λ . Let P x,K denote the orthogonal projection from L (Ω) onto span { h λ : λ ∈ Λ ∩ xK } .Since P λ ∈ Λ ∩ xK h M x f, M λ g i h λ is some approximation of f in P x,K L , the square ofthe distance in (11) is at most k M x f − P x,K f k ≤ k X λ xK h M x f, M λ g i h λ k ≤ C X λ xK |h M x f, M λ g i| = C X λ xK |h f, M x − λ g i| . Set F ( x ) = R Ω f ( ω ) g ( ω ) h ω, x i dω = F − ( f ¯ g )( x − ). Then F ∈ B Ω , and the latterexpression equals C P λ xK | F ( x − λ ) | . If λ xK , then x − λ K , and so weobtain as in the estimate (3) in the proof of Lemma 2 that k M x f − P x,K f k ≤ X λ xK µ G ( U ) Z x − λU | F ♯ ( t ) | dµ G ( t )= X x − λ K µ G ( U ) Z x − λU | F ♯ ( t ) | dµ G ( t ) ≤ µ G ( U ) Z K c U | F ♯ ( t ) | dµ G ( t ) , with U depending only on Λ, but not on x ∈ G . Since F ♯ is in L ( G ), we maychoose K so large that the expression on the right becomes less than ǫ , and thisbound holds uniformly in x .6. Proof of Theorem 1
The proof becomes slightly simpler if we replace Γ byΓ ′ = ( µ b G (Ω) /d Z ) d × { e } × D . This replacement can be made because it is plain that Γ ′ (cid:22) µ b G (Ω)Γ as well as µ b G (Ω)Γ (cid:22) Γ ′ . Thus, by transitivity of the relation ‘ (cid:22) ’, it suffices to prove that ifthe uniformly discrete set Λ is a set of sampling for B Ω , then Γ ′ (cid:22) Λ, and if Λ is aset of interpolation for B Ω , then Λ (cid:22) Γ ′ .The body of the proof is an intermediate step in which we compare Λ with aninteger multiple of one of the quasi-lattices of Lemma 3. Incidentally, this analysisapplies to Γ ′ as well, withΩ ′ := [ − πµ b G (Ω) /d , πµ b G (Ω) /d ] d × { e } × K. This observation will enable us to eliminate the quasi-lattices. In this part ofthe proof, Γ ′ will play a “complementary” role to Λ; Γ ′ is treated as a set ofinterpolation for B Ω ′ when Λ is a set of sampling for B Ω , and vice versa.We begin by covering Ω by an open set Ω such that µ b G (Ω \ Ω) < ǫ /
4. We thentake a neighborhood basis { V } of e in b G and construct the corresponding cubes C V and discrete sets D V ⊆ b G according to Lemma 3. It is easy to see that thecollection S V { d V C V : d V ∈ D V } generates the Borel sets in b G .By taking V small enough, we may choose a cube C = C V and a finite numberof pairwise disjoint translates d j C , d j ∈ D , j = 1 , . . . , N , such thatΩ ∗ = N [ j =1 Ω j ⊆ Ω and µ b G (Ω \ Ω ∗ ) < ǫ µ b G (Ω ∗ ) . This is possible because the Haar measure is regular. We may even assume that N is of the form N = 2 n for a positive integer n because the possibly discrete setof permissible values for µ b G ( C ) is sufficiently dense. More precisely, for arbitrary ANDAU’S NECESSARY DENSITY CONDITIONS 13 c >
1, every interval of the form ( δ, cδ ) will contain a permissible value for µ b G ( C )provided that δ is sufficiently small.By Lemma 3, L ( d j C ) possesses an orthonormal basis { ψ γ : γ ∈ Γ } that islabelled by a quasi-lattice Γ in G . Consequently, L (Ω ∗ ) contains an orthonormalbasis of the form { ψ γ,j , γ ∈ Γ , j = 1 , . . . , N } where ψ γ,j is given explicitly by ψ γ,j ( ω ) = µ b G ( C ) − / h ω, γ i χ d j C ( π ( ω )) for γ ∈ Γ.We now construct another orthonormal basis for L (Ω ∗ ) of the form φ γ,j ( ω ) = µ b G (Ω ∗ ) − / h ω, γ i g j ( ω )for γ ∈ Γ, where g j is a real function such that | g j | = χ Ω ∗ . We obtain g j in thefollowing way. Let U = ( u kl ) , k, l = 1 , . . . , N be a Hadamard matrix, i.e., U hasentries ± N = 2 n .) We set(12) φ γ,j ( ω ) = µ b G (Ω ∗ ) − / h ω, γ i N X k =1 u jk χ d k C ( ω ) . Then { φ γ,j : γ ∈ Γ , j = 1 , . . . , N } is an orthonormal basis for L (Ω ∗ ) with k φ γ,j k ∞ = µ b G (Ω ∗ ) − / . Thusdist L ( φ γ,j , L (Ω)) = k φ γ,j − φ γ,j χ Ω k = k φ γ,j k ∞ k χ Ω ∗ − χ Ω k = µ b G (Ω ∗ ) − / µ b G (Ω ∗ ∆Ω) / < ǫ . Let us first assume that Λ is a set of sampling for B Ω . We then apply thehomogeneous approximation property (Lemma 6) to the frame e λ = M λ χ Ω , λ ∈ Λ,with dual frame h λ , and each of the functions g j χ Ω . We then obtain a compact set K such that dist L ( b G ) (cid:16) M γ g j χ Ω , span { h λ ∈ Λ ∩ γK } (cid:17) < ǫ j = 1 , . . . , N . Therefore,dist L ( b G ) (cid:16) φ γ,j , span { h λ ∈ Λ ∩ γK } (cid:17) < ǫ . This is exactly the hypothesis of Lemma 5, and we have therefore shown that, forevery compact set L , we have(13) (1 − ǫ ) N card (Γ ∩ L ) ≤ card (Λ ∩ KL ) . If Λ is a set of interpolation, we argue similarly. The only difference is that nowthe functions φ γ,j = M γ g j are viewed as a frame, and the functions e λ constitutea Riesz sequence. We apply again the homogeneous approximation property anduse Lemma 4 to get(14) (1 − cǫ )card (Λ ∩ L ) ≤ N card (Γ ∩ KL )for every compact set L , where K is the compact set given by Lemma 6.We have now what we need to finish the proof. To prove part (S) of Theorem 1,we use that Γ ′ is a set of interpolation for B Ω ′ . Hence, by (14), there exists a compact set K ′ such that(15) (1 − cǫ )card (Γ ′ ∩ L ) ≤ N card (Γ ∩ KL )holds for every compact set L ; we may of course adjust ǫ and the approximation ofΩ so that the Γ also suits the approximation of Ω ′ . If Λ is a set of sampling, thencombining (13) with (15), we obtain that(1 − cǫ )card (Γ ′ ∩ L ) ≤ − ǫ card (Λ ∩ K L ) , from which the desired relation Γ ′ (cid:22) Λ follows.Reversing the roles of Λ and Γ ′ , we obtain similarly Λ (cid:22) Γ ′ when Λ is a set ofinterpolation for B Ω .7. Properties of uniform densities
We return to some basic questions about uniform densities that were raised inSection 2.
Lemma 7.
For every uniformly discrete subset Λ of an LCA group G , we have D − (Λ) ≤ D + (Λ) < ∞ .Proof. It is sufficient to prove that both D + (Λ) < ∞ and D − (Λ) < ∞ . Indeed, if D − (Λ) > D + (Λ), then Λ (cid:22) δ Λ for some δ <
1. By the transitivity of the relation‘ (cid:22) ’, this can only happen if D − (Λ) = 0 or D − (Λ) = ∞ .We first prove that D + (Λ) < ∞ . We need to show that there exists a positivenumber α such that Λ (cid:22) α Γ . Let L be a compact subset of Λ. Since Λ is uniformlydiscrete, there is a uniform bound, say M , on the number of points from L ∩ Λ tobe found in each set γK , where K := [ − / , / d × T m × { e } and γ is an elementin Γ . Therefore, card (Λ ∩ L ) ≤ M card (Γ ∩ KL ) , and so Λ (cid:22) M Γ We next prove that D − (Λ) < ∞ . Let us assume that we have α Γ (cid:22) Λ for some α . Then for every positive ǫ there exists a compact set K such that(16) (1 − ǫ ) α card (Γ ∩ L ) ≤ card (Λ ∩ KL )for every compact set L . We may assume that K = B × T m × F , where B is a ballin R d centered at the origin and F is a finite subset of D such F − = F . Then S ∞ n =1 F n is a finitely generated subgroup of D , which has the structure Z l × E with E a finite group. (See [9, p. 451].) To simplify the argument, we may assume that F is just B ′ × E , with B ′ a ball in Z l centered at the origin. We choose L = K n and note that for sufficiently large n we have(17) card (Γ ∩ L ) ≥ (1 − ǫ ) µ G ( L ) . On the other hand, if U ⊆ K is an open set such that the sets λU ( λ in Λ) arepairwise disjoint, we obtain(18) card (Λ ∩ KL ) ≤ µ G ( U ) − µ G ( K n +2 ) ≤ (1 + ǫ ) µ G ( U ) − µ G ( L ) ANDAU’S NECESSARY DENSITY CONDITIONS 15 whenever n is sufficiently large. Combining (16) – (18), we obtain that for ǫ > α ≤ ǫ (1 − ǫ ) µ G ( U ) − , and thus D − (Λ) ≤ µ G ( U ) − .The relation ‘ (cid:22) ’ may be viewed as a relation between discrete measures. Since thecanonical lattice Γ has a highly regular distribution, it should come as no surprisethat we may replace the discrete measure associated with Γ by the Haar measure µ G . Interpreting a uniformly discrete set as a sum of point masses located at thepoints λ of the set, we may generalize the relation ‘ (cid:22) ’ to arbitrary nonnegativemeasures on G . Thus, if ν and τ are two such measures on G , we write ν (cid:22) τ iffor every ǫ > K in G such that(1 − ǫ ) ν ( L ) ≤ τ ( LK )for every compact set L in G . If we set again K = [ − / , / d × T m × { e } , then it is immediate that µ G ( L ) ≤ card (Γ ∩ KL ) and card (Γ ∩ L ) ≤ µ G ( KL )for every compact set L . This implies that µ G (cid:22) Γ and Γ (cid:22) µ G , so that Theorem 1can be restated in the following form. Theorem 1”.
Suppose Λ is a uniformly discrete subset of the LCA group G and Ω is a relatively compact subset of b G . (S) If Λ is a set of sampling for B Ω , then µ b G (Ω) µ G (cid:22) Λ . (I) If Λ is a set of interpolation for B Ω , then Λ (cid:22) µ b G (Ω) µ G . We finally show that, in R d , our uniform densities coincide with the classicalBeurling densities. In R d we use the standard additive notation x + y and K + L instead of the multiplicative notation on arbitrary LCA groups, and we write | U | for the Lebesgue (Haar) measure of U ⊆ R d . Lemma 8. If G = R d , then D − (Λ) = D − B (Λ) and D + (Λ) = D + B (Λ) for everyuniformly discrete set Λ .Proof. Let Λ be a uniformly discrete subset of R d . Then, for every ǫ >
0, thereexists a compact set K = Q R (0) = [ − R/ , R/ d such that(1 − ǫ ) D − (Λ)card ( Z d ∩ L ) ≤ card (cid:0) Λ ∩ ( L + Q R (0)) (cid:1) for every compact set L . Specializing to cubes L = Q h ( y ) , y ∈ R d , we get that(1 − ǫ ) D − (Λ) inf y ∈ R d card ( Z d ∩ Q h ( y ))( h + R ) d ≤ inf y ∈ R d card (Λ ∩ Q h + R ( y ))( h + R ) d . Taking the limit h → ∞ , we obtain (1 − ǫ ) D − (Λ) ≤ D − B (Λ), and so D − (Λ) ≤ D − B (Λ)since the inequality holds for every positive ǫ . Conversely, we may for any given ǫ > h > ∩ Q h ( y )) h d ≥ (1 − ǫ ) D − B (Λ)for every point y in R d and h > h . Now partition R d into cubes Q h ( hk ) , k ∈ Z d , whose interiors are disjoint. Given a compact set L ⊆ R d , there exist finitely many k j ∈ Z d , j = 1 , . . . J , such that L ⊂ J [ j =1 Q h ( hk j ) ⊂ L + Q h (0) . Then by (19)card (cid:16) Λ ∩ ( L + Q h (0)) (cid:17) ≥ J X j =1 card (Λ ∩ Q h ( hk j )) ≥ h d (1 − ǫ ) J D − B (Λ) . Since ( h + 1) d ≥ card ( Z d ∩ Q h ( hk j )), it follows thatcard (cid:0) Λ ∩ ( L + Q h (0)) (cid:1) ≥ (1 − ǫ ) (1 + 1 /h ) − d D − B (Λ) card ( Z d ∩ L ) . We may choose ǫ arbitrarily small and h arbitrarily large, and hence D − (Λ) ≥D − B (Λ).The identity D + (Λ) = D + B (Λ) is proved similarly.8. Arbitrary LCA Groups
So far we have assumed that b G is compactly generated. This is not a seriousrestriction, as shown by the following lemma. (See also [5].) Lemma 9.
Assume that Ω ⊆ b G is relatively compact and let H be the open subgroupgenerated by Ω ⊆ b G . Then H is compactly generated and there exists a compactsubgroup K ⊆ G such that every f ∈ B Ω is K -periodic.Furthermore, the quotient G/K factors as
G/K ≃ R d × T k × D for some count-able discrete abelian group D and ( G/K ) b = H , where H is the open subgroup of b G that is generated by the spectrum Ω .Proof. Choose an open, relatively compact neighborhood V of the spectrum Ω ⊆ b G ,and let H be the open subgroup of b G that is generated by V . Then b G/H is discrete,and thus the group (cid:0) b
G/H (cid:1)b is compact. We claim that K := H ⊥ is the subgroupwe are looking for. Let f ∈ B Ω , x ∈ G, k ∈ K , then by the inversion formula f ( xk ) = Z Ω b f ( ω ) ω ( xk ) dµ b G ( ω )= Z Ω b f ( ω ) ω ( x ) ω ( k ) dµ b G ( ω )= Z Ω b f ( ω ) ω ( x ) dµ b G ( ω ) = f ( x )since k ∈ H ⊥ and Ω ⊆ H . ANDAU’S NECESSARY DENSITY CONDITIONS 17
Since H is compactly generated, H is isomorphic to a group H ≃ R d × Z k × L for some compact group L by the structure theorem for LCA groups [9, Thm. 9.8].Consequently, b H ≃ bb G/H ⊥ ≃ G/K ≃ R d × T k × D , where D = b L is a discrete group.Consequently, every bandlimited function f ∈ B Ω lives on a quotient G/K andmay be identified with a function ˜ f ∈ L ( G/K ). Example . Let Q p be the group of p -adic numbers [9] with dual group isomorphicto Q p . The p -adic numbers possess a “quasi-metric” | · | p such that | x + y | p ≤ max( | x | p , | y | p ) for all x, y ∈ Q p . Moreover, for each n ∈ Z , K n := { x ∈ Q p : | x | p ≤ n } is a compact-open subgroup of Q p . As a consequence, every relatively compactset Ω ⊆ Q p generates a compact group H contained in some K n . In particular, Q p does not contain any lattice.It seems that our main theorem does not say anything about sampling in p -adicgroups. However, Lemma 9 says that we may assume without loss of generality that b G is one of the K n ’s where K n contains the group H generated by the spectrum Ω.Furthermore, all functions in B Ω are H ⊥ -periodic and thus live on the discrete group Q p /H ⊥ . Thus we may apply Theorem 1 to the pair G = Q p /H ⊥ and H ⊆ K n .9. Closing remarks ( ) In his paper [12], Landau made a slightly weaker assumption on Ω whenconsidering sets of sampling. Instead of taking Ω to be relatively compact, heassumed that Ω had positive measure. It is clear that we may similarly take Ωto have positive Haar measure in part (S) of Theorem 1 because such Ω can beapproximated by compact sets contained in Ω. Note that this relaxation cannotbe made in part (I) of Theorem 1.( ) Landau used his results in [12] to prove a conjecture of A. Beurling concerningthe lower uniform density of sets in R d for which so-called balayage is possible. Wedo not wish to go into detail about Beurling’s problem, but we would like to pointout that, using our notion of density, we may extend Landau’s result concerningbalayage. The restriction we have to make is that the group G be of the form G = R d × Z m × K with d ≥
1. Theorem 5 in [12] extends from the setting of R d to such groups, under the same regularity conditions on the spectrum. Thedetails needed to carry out this extension can be found in [2, pp. 341–350] and inLandau’s paper [12].( ) In his thesis [14], Marzo proved that for every relatively compact set Ω in R d we can find sets of sampling and sets of interpolation for B Ω of Beurling densitiesarbitrarily close to those given by Landau’s theorem. It would be interesting toknow if, similarly, our density conditions are optimal for every relatively compactset in a general LCA group. ( ) In section 2, we excluded the case of compact groups. Our result is certainlyof no interest for compact groups, but for such groups one can state closely relatedand nontrivial problems. An example is the recent work of J. Ortega-Cerd`a andJ. Saludes on Marcinkiewicz-Zygmund inequalities [15]. Their work deals with thegroup G = T and the asymptotic behavior of sets of sampling and interpolationwhen the size of the spectrum grows and we require uniform bounds on the norms.Another, probably much more difficult problem, is to describe similarly asymptoticdensity conditions when G = T m and both the spectrum and m grow. References [1] R. Balan, P. G. Casazza, C. Heil, and Z. Landau,
Density, overcompleteness, and localizationof frames. I. Theory , J. Fourier Anal. Appl. (2006), 105–143.[2] A. Beurling, The Collected Works of Arne Beurling, Vol. 2 Harmonic Analysis , Birkh¨auser,Boston, 1989.[3] O. Christensen, B. Deng, and C. Heil,
Density of Gabor frames , Appl. Comput. Harmon.Anal. (1999), 292–304.[4] G. B. Folland, A Course in Abstract Harmonic Analysis , CRC Press, Boca Raton, 1995.[5] H. G. Feichtinger, K. Gr¨ochenig,
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On Landau’s necessary density conditions for samplingand interpolation of band-limited functions , J. London Math. Soc. (2) (1996), 557–565.[7] C. Heil, History and evolution of the density theorem for Gabor frames,
J. Fourier Anal.Appl. (2007), 113–166.[8] C. Heil and G. Kutyniok, The homogeneous approximation property for wavelet frames , J.Approx. Theory (2007), 28–46.[9] E. Hewitt and K. A. Ross,
Abstract Harmonic Analysis I , Springer-Verlag, Berlin/Heidel-berg/New York, 1963.[10] I. Kluv´anek,
Sampling theorem in abstract harmonic analysis , Mat.-Fyz. ˇCasopis Sloven.Akad. Vied , (1965), 43–48.[11] G. Kutyniok, Affine Density in Wavelet Analysis , Lecture Notes in Mathematics ,Springer-Verlag, Berlin, 2007.[12] H. J. Landau,
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Marcinkiewicz–Zygmund inequalities , J. Approx. Theory (2007), 237–252.[16] J. Ortega-Cerd`a and K. Seip,
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On the connection between exponential bases and certain related sequences in L ( − π, π ), J. Funct. Anal. (1995), 131–160. ANDAU’S NECESSARY DENSITY CONDITIONS 19
Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, A-1090 Vi-enna, Austria
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