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APPEARED IN BULLETIN OF THE

arXiv:math/9204238v1 [math.CV] 1 Apr 1992APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 26, Number 2, April 1992, Pages 322-328DENSITY THEOREMS FOR SAMPLING AND INTERPOLATIONIN THE BARGMANN-FOCK SPACEKRISTIAN SEIPAbstract. We give a complete description of sampling and interpolation inthe Bargmann-Fock space, based on a density concept of Beurling.

Roughlyspeaking, a discrete set is a set of sampling if and only if its density in everypart of the plane is strictly larger than that of the von Neumann lattice, andsimilarly, a discrete set is a set of interpolation if and only if its density in everypart of the plane is strictly smaller than that of the von Neumann lattice.1. Introduction and resultsThe work presented in this announcement is based on Beurling’s lectures onbalayage of Fourier-Stieltjes transforms and interpolation for an interval on R1 [3].We observe that Beurling’s problems concerning functions of exponential type havenatural counterparts for functions of order two, finite type and find that, indeed, sohave his main results.

The most interesting part, however, is that Beurling’s ideasare applicable also in the Hilbert space setting, yielding a complete description ofsampling and interpolation in the Bargmann-Fock space. The simplicity of theseresults is quite remarkable when compared to the situation in the Paley-Wienerspace (the corresponding Hilbert space of functions of exponential type) and to theextensive literature on nonharmonic Fourier series and, in particular, Riesz basesof complex exponentials [20].This research is motivated by a recent development in signal analysis and appliedmathematics, which was initiated by Daubechies, Grossmann, and Meyer [5, 4, 6].Their work inspired us to search for a general characterization of the informationneeded to represent signals, as functions in the Bargmann-Fock space.

Our resultscan be seen as sharp statements about the Nyquist density and its meaning in thiscontext.In order to describe more precisely the problems to be considered, a few defini-tions are needed. For α > 0, let dµα(z) = (α/π)e−α|z|2dxdy, z = x + iy, and definethe Bargmann-Fock space F 2α to be the collection of entire functions f(z) for which∥f∥2 = ∥f∥α,2 =ZC|f(z)|2dµα(z) < ∞.Received by the editors August 9, 1991 and, in revised form, November 19, 1991.1991 Mathematics Subject Classification.

Primary 30D10, 30E05, 46E20; Secondary 81D30.c⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1

2KRISTIAN SEIPF 2α is a Hilbert space with reproducing kernel K(z, ζ) = eαzζ; i.e., for every f ∈F 2αwe havef(z) = ⟨f, K(z, ·)⟩=ZCf(ζ)K(z, ζ) dµα(ζ).The normalized reproducing kernels, kζ(z) = K(ζ, ζ)−1/2K(ζ, z), can be view-ed as the natural (well-localized) building blocks of F 2α. They correspond, via theBargmann transform, to the canonical coherent states of quantum mechanics andto Gabor wavelets in signal analysis.

This relation is the reason for the impor-tance of the Bargmann-Fock space; see [8] for general information and [5] for morebackground on the problems treated here.We say that a discrete set Γ of complex numbers is a set of sampling for F 2α ifthere exist positive numbers A and B such thatA∥f∥22 ≤Xz∈Γe−α|z|2|f(z)|2 ≤B∥f∥22(1)for all f ∈F 2α. If to every l2-sequence {aj} of complex numbers there exists anf ∈F 2α such that e−α|zj|2/2f(zj) = aj for all j, then Γ = {zj} is said to be a set ofinterpolation for F 2α.

A set of sampling corresponds, in the terminology of [7], toa frame of coherent states. A set of both sampling and interpolation (which doesnot exist) would correspond to a Riesz basis of coherent states.With a view to applications in physics and signal analysis, Daubechies and Gross-mann posed the problem of finding the lattices zmn = ma + inb, m, n ∈Z, that aresets of sampling [5].

They proved that a lattice could be a set of sampling only ifab < π/α and conjectured this condition also to be sufficient. For ab = π/(αN),N an integer ≥2, they found (1) to hold by providing explicit expressions for theoptimal constants A, B. Daubechies was later able to show that a lattice is a set ofsampling whenever N −1 < 0.996 [4].We prove that the density criterion of the Daubechies-Grossmann conjectureapplies not only to lattices, but to arbitrary discrete sets.

We should add here thatthe conjecture was proved independently by Lyubarskii [12] and by Wallst´en andthe author [19].For the description to be given of sets of sampling and interpolation, we needBeurling’s density concept as generalized by Landau [11]. We consider then uni-formly discrete sets, i.e., discrete sets Γ = {zj} for which q = infj̸=k |zj −zk|> 0.

We fix a compact set I of measure 1 in the complex plane, whose boundaryhas measure 0. Let n−(r) and n+(r) denote, respectively, the smallest and largestnumber of points from Γ to be found in a translate of rI.

We define the lower andupper uniform densities of Γ to beD−(Γ) = lim infr→∞n−(r)r2andD+(Γ) = lim supr→∞n+(r)r2,respectively. It was proved by Landau that these limits are independent of I.Our main theorems are the following (the sufficiency part of the theorems wereobtained in collaboration with Wallst´en [19].Theorem 1.1.

A discrete set Γ is a set of sampling for F 2α if and only if it canbe expressed as a finite union of uniformly discrete sets and contains a uniformlydiscrete subset Γ′ for which D−(Γ′) > α/π.

SAMPLING AND INTERPOLATION IN THE BARGMANN-FOCK SPACES3Theorem 1.2. A discrete set Γ is a set of interpolation for F 2α if and only if it isuniformly discrete and D+(Γ) < α/π.Remark 1.

Decomposition and interpolation theorems for general discrete setswere obtained in [10], however, without any indication of a critical density. Theresults in [10] appear as part of a certain trend in harmonic analysis, and theanalogy to the theory of nonharmonic Fourier series does not seem to have beenrealized.Remark 2.

The lattice with a = b =pπ/α is called the von Neumann lattice,since von Neumann claimed (without proof) that it is a set of uniqueness [13]; manyproofs have later been given [2, 14, 1, 19]. See [9] for an attempted repair of the“defect” of the von Neumann lattice that it is neither a set of sampling nor one ofinterpolation.We consider also the analogues in our setting of the problems treated in [3].

Weintroduce then the Banach space F ∞α , consisting of those entire functions f(z) forwhich∥f∥∞= ∥f∥α,∞= supz e−α|z|2/2|f(z)| < ∞.Γ is said to be a set of sampling for F ∞αif there exists a positive number K suchthat∥f∥∞≤K supz∈Γe−α|z|2/2|f(z)|for all f ∈F ∞α . If to every bounded sequence {aj} of complex numbers there existsan f ∈F ∞αsuch that e−α|zj|2/2f(zj) = aj for all j, we say that Γ = {zj} is a setof interpolation for F ∞α .

We have then the following counterparts of Beurling’s twodensity theorems in [3] (we are using the term sampling instead of balayage as in[3], which seems natural since we no longer have the relation to Fourier-Stieltjestransforms).Theorem 1.3. A discrete set Γ is a set of sampling for F ∞α if and only if it containsa uniformly discrete subset Γ′ for which D−(Γ′) > α/π.Theorem 1.4.

A discrete set Γ is a set of interpolation for F ∞αif and only if it isuniformly discrete and D+(Γ) < α/π.Let us remark, as Beurling did, that the problems and some of the results extendto several variables. We would also like to mention the following interesting ques-tion: What are the corresponding density theorems for weighted Bergman spaces?See [16, 18] for a treatment of this problem.2.

The necessity parts of the theorems—indication of proofIn this section we make a few remarks to indicate how to prove the necessityparts of the theorems. Details are given in [17].

When unspecified, p is taken to beeither 2 or ∞.We remark first that the translations(Taf )(z) = eαaz−α|a|2/2f(z −a)

4KRISTIAN SEIPact isometrically in F pα. This translation invariance implies immediately that Γ+z isa set of sampling (interpolation) if and only if Γ is a set of sampling (interpolation)and it permits us to translate our analysis around an arbitrary point z to 0.Another important feature of F pα is the following compactness property: If {fn}is a sequence in the ball{f ∈F pα : ∥f∥p ≤R},then there is a subsequence {fnk} converging pointwise and uniformly on compactsets to some function in the ball.

This is immediate from the definition of F pα anda normal family argument.Following Beurling, for a closed set Γ, we let W(Γ) denote the collection of weaklimits of translates Γ+z [3, p. 344]. The compactness property and the translationinvariance of F pα make W(Γ) a crucial tool in our analysis.

Indeed, it turns out thatall of Beurling’s arguments concerning W(Γ) can be carried over to our situation.Most of the work needed to prove the necessity parts of Theorems 1.3 and 1.4 con-sists in transferring Beurling’s arguments. In addition to the ingredients mentionedabove, a simple substitute for Bernstein’s theorem is used.

Moreover, adapting anidea of Landau [11], we make use, at a certain stage in the proof of Theorem 1.3,of the nice properties of the normalized monomials (normalized in F 2α), see [8, p.39; 15].For the L2 problem, the basic auxiliary result is the following lemma.Lemma 2.1. There is no discrete subset of C that is both a set of sampling and aset of interpolation for F 2α.This lemma has the following consequences.Lemma 2.2.

If Γ is a set of sampling for F 2α, then so is Γ \ {ζ} for any ζ ∈Γ.Lemma 2.3. If Γ is a set of interpolation for F 2α, then so is Γ∪{ζ} for any ζ ̸∈Γ.The main difficulty in proving the necessity part of Theorem 1.1 consists inshowing that D−(Γ) > α/π if Γ is uniformly discrete and a set of sampling.

Thisproblem can now be dealt with in the following way. Consider such a Γ.

It is easyto show that W(Γ) consists only of sets of sampling. By Lemma 2.2 we have thatevery set of sampling for F 2α is a set of uniqueness for F ∞α .

For suppose Γ0 is a setof sampling for F 2α and that g ∈F ∞αvanishes on Γ0. Then the functionf(z) = g(z)/(z −z1)(z −z2),z1, z2 ∈Γ0, belongs to F 2α and vanishes on Γ0\{z1, z2}.

This contradicts Lemma 2.2.Thus every set in W(Γ) is a set of uniqueness for F ∞α . It can be proved that Γis a set of sampling for F ∞αif and only if every Γ0 ∈W(Γ) is a set of uniquenessfor F ∞α(see Theorem 3 in [3, p. 345]).

Hence by Theorem 1.3, D−(Γ) > α/π.As to the necessity part of Theorem 1.2, we remark that Lemma 2.3 enables usto carry over Beurling’s technique used for the corresponding L∞problem; here aslight modification of the key notion ‘ρ(z; Γ)’ is needed, see [3, p. 352].

SAMPLING AND INTERPOLATION IN THE BARGMANN-FOCK SPACES53. The sufficiency parts of the theorems—indication of proofDetails can be found in [19].Let Λ = {λmn} denote a square lattice; that is, λmn =pπ/α (m + in) for allintegers m, n and some positive number α. α/π will be referred to as the density ofΛ.

We observe that the Weierstrass σ-function, σ(z), associated to Λ plays a role inthe Bargmann-Fock space analogous to that of the sine in the Paley-Wiener space.This permits us to use techniques similar to some of those employed for functionsof exponential type.We introduce the analogues of σ(z) for uniformly discrete sets that are close toa square lattice in the following sense. Γ = {zmn} is uniformly close to Λ if thereexists a positive number Q such that |zmn −λmn| ≤Q for all m and n.To Γ, uniformly close to a square lattice, we associate a function g(z) defined byg(z) = (z −z00)Ym,n′ 1 −zzmnexp zzmn+ 12z2λ2mn(2)where z00 is the point of Γ closest to 0.

Using the quasi-periodicity of the σ-function,we obtain the following estimates on the growth of g.Lemma 3.1. Let Γ be uniformly close to the square lattice Λ of density α/π.

Thenthere exist constants C1, C2 and c, depending only on Q and q, such that for everyz we have|e−α|z|2/2g(z)| ≥C1 e−c|z| log |z| dist(z, Γ),|e−α|z|2/2g(z)| ≤C2 ec|z| log |z|,and for every zmn ∈Γ we have|e−α|zmn|2/2g′(zmn)| ≥C1 e−c|zmn| log |zmn|.By this lemma and the calculus of residues, we obtain the following Lagrange-type interpolation formula.Lemma 3.2. Let Γ = {zmn} be uniformly close to the square lattice of densityβ/π, and let g be the function associated to Γ by ( 2).

If α < β we have for eachf ∈F ∞αf(z) =Xm,nf(zmn)g′(zmn)g(z)z −zmnwith uniform convergence on compact sets.The difficulty in proving the sufficiency part of Theorem 1.1 consists in verifyingthe left inequality in (1). We assume then, without loss of generality, that Γ isuniformly discrete and uniformly close to the square lattice of density β/π, whereD−(Γ) = β/π; see [3, p. 356] for an argument justifying this claim.

We writeZC|f(z)|2 dµα(z) =Xk,lZR|(Tλklf)(z)|2 dµα(z)

6KRISTIAN SEIPwhere R = {z = x + iy : |x| < 12p1/α, |y| < 12p1/α}. In order to estimate thesummands on the right, we use Lemma 3.2 to write(Tλklf)(z) =Xm,n(Tλklf)(zmn + λkl)g′λkl(zmn + λkl)gλkl(z)z −zmn −λkl,where gλkl is the function associated to Γ+λkl by (2).

Lemma 3.1 is used to estimatethis expression, and after some computation we obtain the desired estimate.The sufficiency part of Theorem 1.3 can be proved in the same way, or moreeasily, by Beurling’s method [3, p. 346].In order to prove the sufficiency parts of Theorems 1.2 and 1.4, we note first thatwe may assume that Γ is uniformly close to a square lattice of density β/π, whereD+(Γ) = β/π [3, p. 356]. The interpolation problem is then solved explicitly bythe following formula,f(z) =Xm,namneαzmnz−α|zmn|2 g−zmn(z −zmn)(z −zmn).Lemma 3.1 is used to verify this assertion.References1.

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Soc. (to appear).17.Density theorems for sampling and interpolation in the Bargmann-Fock space I, J.Reine Angew.

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SAMPLING AND INTERPOLATION IN THE BARGMANN-FOCK SPACES719. K. Seip and R. Wallst´en, Density theorems for sampling and interpolation in the Bargmann-Fock space II, J. Reine Angew.

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