Weak conditions for interpolation in holomorphic spaces

Publicacions Matem`atiques, Vol. 44 (2000), 277–293 WEAK CONDITIONS FOR INTERPOLATION IN HOLOMORPHIC SPACES Alexander P. Schuster and Kristian Seip Abstract An analogue of the notion of uniformly separated sequences, expressed in terms of extremal functions, yields a necessary and sufficient condition for interpolation in Lpspaces of holomorphic functions of Paley-Wiener-type when 0 <p≤1, of Fock-type when 0<p≤2, and of Bergman-type when 0 <p<∞. Moreover, if a uniformly discrete sequence has a certain uniform non-uniqueness property with respect to any such Lpspace (0 <p<∞), then it is an interpolation sequence for that space. The proofs of these results are based on an approximation theorem for subharmonic functions, Beurling’s results concerning compactwise limits of sequences, and the description of interpolation sequences in terms of Beurling-type densities. Details are carried out only for Fock spaces, which represent the most difficult case. 1. Introduction This paper studies some consequences of the fact that interpolation sequences for Bergman, Fock, and Paley-Wiener spaces are invariant under certain natural group actions and corresponding compactwise limits. We will show how and when this invariance implies that the defining property of interpolation sequences is equivalent to apparently weaker statements about sequences. Two such weak conditions will be considered: The first is the analogue of Carleson’s condition of uniform separation (cf. [11]); the second is a condition of “uniform non-uniqueness”, in some sense dual to a condition of Beurling used to describe sampling sequences. Our methods apply to all of the spaces mentioned above, but the main focus will be put on Fock spaces, because they represent the most difficult case. We begin by describing our results in that setting. 1991 Mathematics Subject Classification. 30H05, 46E15. 278 A. P. Schuster, K. Seip Let dσ denote Lebesgue area measure on C. We define fp α,p =C |f(z)|pe−pα|z|2dσ(z) for α>0 and p<∞, and fα,∞= supz|f(z)|e−α|z|2. The Fock space Fp αconsists of those entire functions fsuch that fα,p <∞.Of basic importance is that the translation operator Tζ,α (ζ∈C) defined as Tζ,αf(z)=f(z−ζ)eα(2ζz−|ζ|2) acts isometrically on Fp α. This fact will sometimes be referred to as the translation invariance of Fp α. We say that a sequence Γ = {γn}of distinct points in Cis interpolating for Fp αif for every sequence {an}satisfying  n |an|pe−pα|γn|2<∞, there is an f∈Fp αsuch that f(γn)=anfor all n. For the case p=∞, we require the existence of an f∈F∞ αwith f(γn)=anwhenever sup n |an|e−α|γn|2<∞. For a sequence Γ = {γn}of distinct points, set Γk={γn}n:n=kand let Fp α(Γk) denote the closed subspace of Fp αconsisting of functions vanishing on Γk. (We will keep this notation when Fp αis replaced by other spaces of holomorphic functions.) Our first main theorem is the following: Theorem 1. Let Γ={γn}be a sequence of distinct points in C, and suppose 0<p≤2. Then Γis interpolating for Fp αif and only if there is aδ>0such that sup{|f(γk)|:f∈Fp α(Γk),fα,p ≤1}≥δeα|γk|2for all k.(1) A normal family argument shows that the supremum on the lefthand side of (1) is attained, and thus it is in fact a maximum. A simple example (see Section 3) shows that Theorem 1 fails for p>2. We note that condition (1) is indeed “apparently weaker” than the condition of being interpolating, because (1) says only that we can solve the interpolation problems f(γk)=1,f(γn) = 0 for n=k, with control of the norms of the solutions. The fact that we can achieve such norm control when Γ is interpolating follows from the open mapping theorem in a standard way. Thus the necessity of (1) for Γ to be interpolating is trivial, modulo the open mapping theorem. Interpolation in holomorphic spaces 279 It is instructive to restate and discuss conditon (1) in a more general setting. If Bis a normed or quasi-normed linear space of holomorphic functions defined on some domain Ω, we say that a sequence Γ of distinct points from Ω is a weak interpolation sequence for Bif there exists a δ>0 such that (2) sup{|f(γk)|:f∈B(Γk),fB≤1} ≥δsup{|f(γk)|:f∈B,fB≤1} for all k. Thus (1) says that Γ is a weak interpolation sequence for Fp α. The connection between interpolation and weak interpolation is part of Carleson’s classical interpolation theorem (cf. Chapter IX of [3]): A sequence is interpolating for Hpif and only if it is a weak interpolation sequence for Hp, where Hpdenotes the Hardy space of the open unit disk Dof Cand 0 <p≤∞. (In this case, interpolating Blaschke products are extremal functions for the left-hand side of (2).) In [11], we showed that the same statement is true for the Bergman spaces Ap β of the unit disk (see Section 4 for definition) when 0 <p<∞. (The proof technique of [11] is different from the one used here and does not carry over to the Fock and Paley-Wiener spaces.) In Section 4, we show that for the Paley-Wiener spaces PWp τ, this characterization holds only when 0 <p≤1. The p-dependence in these results is quite curious. It can be attributed directly to the underlying geometry of our spaces, respectively to the line (Paley-Wiener), the plane (Fock), and the disk (Bergman). However, in general, it is less clear why in some cases weak interpolation implies interpolation (like for the Hardy spaces), and in some it does not (like for the Dirichlet space; see Section 5). Theorem 1 will be obtained as a consequence of a closely related result, which we will now describe. For this we need Beurling’s notion of compactwise limits. A sequence Qjof closed sets converges strongly to Q, denoted Qj→Q,if[Q, Qj]→0; here [Q, R] denotes the Fr´echet distance between two closed sets Qand R, i.e., [Q, R] is the smallest number tsuch that Q⊂{z:d(z,R)≤t}and R⊂{z:d(z,Q)≤t}, where d(·,·) denotes Euclidean distance in C.Qjconverges compactwise to Q, denoted QjQ, if for every compact set D,(Qj∩D)∪∂D → (Q∩D)∪∂D. For a closed set Γ, we let W(Γ) denote the collection of sets Γsuch that Γ+ajΓfor some sequence {aj}. If Γ is interpolating for Fp α, then using the translation invariance of Fp αand a normal family argument, we see that all sequences in W(Γ) are also interpolating for Fp α. 280 A. P. Schuster, K. Seip A sequence Γ is a uniqueness sequence for Fp αif the only element of Fp αvanishing on Γ is the zero function, and otherwise we say that Γisanon-uniqueness sequence for Fp α. A sequence is called uniformly discrete if the infimum of the Euclidean distances between distinct points is strictly positive. With this terminology, our second main theorem can be stated as follows: Theorem 2. Let Γ={γn}be a sequence of distinct points in Cand suppose p<∞. Then Γis interpolating for Fp αif and only if Γis uniformly discrete and every Γ∈W(Γ) is a non-uniqueness sequence for Fp α. This uniform non-uniqueness conditon is known to be necessary for a sequence to be interpolating for Fp α(see Section 2), and so the problem is to prove that this apparently rather weak condition implies that a sequence is interpolating. Our proof of this implication is easily transferred to Paley-Wiener and Bergman spaces, and in fact to all the weighted spaces described in [15]. Theorem 2 appears particularly interesting when contrasting it with the Hpcase: The analogue of Theorem 2 fails for Hp, as can be seen by constructing a suitable “uniform” Blaschke sequence which does not meet the Carleson condition. The next two sections contain the proofs of Theorem 2 and Theorem 1. In Section 4, we will summarize our findings in the Paley-Wiener and Bergman spaces, with only indications of proofs. Finally, in Section 5, the general question about the relationship between interpolation and weak interpolation is discussed briefly in the case that Bis a Hilbert space. In particular, we show that weak interpolation does not imply interpolation in the Dirichlet space. In what follows, we write fgwhenever there is a constant Ksuch that f≤Kg, and f≃gif both fgand gf. 2. Proof of Theorem 2 We comment first on the necessity of the two conditions of Theorem 2. The easy proof that an interpolation sequence is uniformly discrete can be found in [9] (cf. Proposition 3). The proof of the fact that the uniform non-uniqueness condition necessarily holds can be found in that same paper for p≥1 (cf. Theorem 2). The case p≤2 follows from Lemma 6.2 of [12] with L2replaced by Lp,p≤2. (Cf. also the proof of Theorem 2 in Section 3.) We describe next the basic ingredients for the proof of the converse implication. Interpolation in holomorphic spaces 281 The sequence Γ = {γn}of distinct points in Cis a sampling sequence for Fp αif fα,p ≃{f(γn)e−α|γn|2}p for f∈Fp α. We will need the description of sampling and interpolation sequences in terms of lower and upper uniform densities. To this end, fix Γ and denote by n(z,r) the number of points of the sequence Γ in D(z,r), which is the disk of centre zand radius r. The lower uniform density of Γ is defined as D−(Γ) = lim inf r→∞ min z∈C n(z,r) πr2, and the upper uniform density of Γis D+(Γ) = lim sup r→∞ max z∈C n(z,r) πr2. Note that 0 ≤D−(Γ) ≤D+(Γ) <∞when Γ is uniformly discrete. We shall need the following theorem: Theorem A. Suppose Γis uniformly discrete and 0<p≤∞. Then Γ is sampling for Fp αif and only if D−(Γ) >(2α)/π, and Γis interpolating for Fp αif and only if D+(Γ) <(2α)/π. The results for p= 2 and p=∞are proved in [12] and [16]; the methods of those papers extend directly to the case p≤2 and with minor modifications to the case 2 <p<∞. A different approach for the case p≥1 can be found in [9]. (The “sampling part” of that paper contains a proof valid for all p.) Note also that in fact Theorem 1 gives an interesting new proof of the necessity of the density condition for interpolation when p≤2. The second ingredient is a result from [13], which is an analogue of a theorem of Beurling [2]. This theorem, which reflects the translation invariance of sampling sequences, is crucial for proving the sampling part of Theorem A. Theorem B. Suppose Γis uniformly discrete. Then Γis sampling for F∞ αif and only if every Γ∈W(Γ) is a uniqueness sequence for F∞ α. The third auxiliary result is a special case of an interesting approximation principle of Lyubarskii and Malinnikova [5]. It states that any subharmonic function can be approximated by the logarithm of the modulus of an entire function outside a “small” exceptional set. We need the following special version of it, in essence proved earlier by Lyubarskii and Sodin [7] and stated as Theorem 3 in [5]: 282 A. P. Schuster, K. Seip Theorem C. Suppose φis a subharmonic function in Csuch that ∆φ≃ 1, where ∆is the Laplacian ∂2 ∂∂ . Then there exists an entire function G, with uniformly discrete zero sequence Γ, such that |G(z)|≃eφ(z)d(z,Λ).(3) Suppose now that Γ is uniformly discrete and W(Γ) contains only non-uniqueness sequences for Fp α. Let us first explain informally the plan of our proof. We will use Theorem C to construct another sequence Λ, which “completes” Γ in the sense that the union of Γ and Λ has a uniform distribution just like a lattice. We shall then transform the problem in such a way that it is solved by applying Theorem B to the “dual” sequence Λ and an appropriate space F∞ α. We turn to the details. Let Fbe some entire function vanishing precisely on Γ. We claim that for any β>πD +(Γ)/2 we can find a uniformly discrete sequence Λ and an entire function Gvanishing on Λ such that |F(z)G(z)|≃d(z,Γ)d(z,Λ)eβ|z|2.(4) (Thus the zeros of FG are distributed in the same regular way as the points in a lattice of density (2β)/π.) This is done in the following fashion (cf. [1, pp. 113–114]). Set χr(z)=1/(πr2),|z|<r 0,|z|≥r, and let ν= n δγn be the measure consisting of a point mass at each point of our sequence Γ. Choose rso big that 2β−πν ∗χr(z)≥), where )=β−πD+(Γ)/2 and ν∗χrdenotes the convolution of νand χrover C. We set v=(ν−ν∗χr)∗E, where E= log |z|. Actually, to be precise, this function is defined as v(z) = limt→∞ vt(z), where vtcorresponds to the finite sequence consisting of those γnwhich satisfy |γn|<t. The function vtis clearly well-defined, and the limit exists because vt(z)=vτ(z)if|z|+r<t<τ, by the mean value property for harmonic functions. We define φ(z)=β|z|2+v(z)−log |F(z)| Interpolation in holomorphic spaces 283 and observe that )≤∆φ(z)≤β. It remains to apply Theorem C and use the estimate |v(z)−log d(z,Γ)|1, which follows from the definition of v. It is essential to note that Λ = {λn}is constructed in such a way that we obtain the identity D+(Γ) = (2β)/π −D−(Λ). Namely, the Riesz measure ∆φ(z)dσ(z)ofφis atomized by splitting the plane into “cells”, each of mass 1 with respect to (2/π)∆φ(z)dσ(z), and placing one λnin each “cell”; we refer to [5] for details. To show that Γ is an interpolation sequence for Fp α, it therefore suffices to prove that D−(Λ) >2(β−α)/π. In view of Theorem A, we need only show that Λ is a sampling sequence for F∞ β−αand by Theorem B, this reduces to showing that W(Λ) contains only uniqueness sequences for F∞ β−α. Suppose then that Λ∈W(Λ). Recall that it is obtained as a compactwise limit of a sequence Λ + ak. Choose a subsequence of {ak},say {akj}, such that also Γ + akjΓ∈W(Γ). By a normal family argument applied to the functions Takj,β(FG), it is clear that there exist functions ˜ Fand ˜ Gvanishing respectively on Γand Λsuch that |˜ F(z)˜ G(z)|≃d(z,Γ)d(z,Λ)eβ|z|2.(5) This proves that Γ∪Λis a uniqueness sequence for Fp β, because otherwise there would have to exist an entire function fsuch that f˜ F˜ G∈Fp β. However, using (5) and the subharmonicity of |f|pin small disks around each point in Γand Λ, we obtain f˜ F˜ Gp β,p ≃C |f(z)|pdσ(z)<∞, which is a contradiction unless f≡0. On the other hand, since Γis a non-uniqueness sequence for Fp α, there is a nontrivial function g∈Fp α which vanishes on Γ.IfΛ is likewise a non-uniqueness sequence for F∞ β−α, there exists a function h∈F∞ β−αvanishing on Λsuch that gh ∈ Fp β, which we have seen is impossible. So Λis a uniqueness sequence for F∞ β−α, and the proof of Theorem 2 is complete. 284 A. P. Schuster, K. Seip Note that an application of Theorem C with φ(z)=α|z|2shows that Theorem 2 fails when p=∞. Since the sequence Λ satisfies (3), it is a non-uniqueness sequence for F∞ α. A normal family argument shows that the same is true of every member of W(Λ). On the other hand, it is clear by construction that D+(Λ) = (2α)/π, so that by Theorem A, Λ is not interpolating. Theorem 2 does have an analogue for p=∞, however, if one considers instead the space F∞,0 α, which consists of those entire functions satisfying |f(z)|eα|z|2→0as|z|→∞. In this setting, Γ is interpolating if the problem f(γn)=anis solvable whenever aneα|γn|2→0asn→∞. The statement and proof of the result are then identical to that of Theorem 2, with Fp αreplaced by F∞,0 α. 3. Proof of Theorem 1 Several remarks are in order before we turn to the proof of the sufficiency of the weak interpolation condition of Theorem 1. First, note that when 0 <p≤1, one can solve the interpolation problem explicitly: f(z)= k ak Gk(z) Gk(γk), where Gkis an extremal function of the left-hand side of (1). It is clear that this series converges locally uniformly to a function in Fp αand solves the interpolation problem when the ak’s satisfy the compatibility condition. Second, since Fp αis continuously embedded into F2 αfor p≤2, and since the interpolation sequences are independent of p, it suffices to prove the theorem for the case p= 2, which is what we will do. Third, we note that the theorem fails when p>2. To show this, we again appeal to Theorem C to find an entire G, with uniformly discrete zero sequence Γ, such that G(z)≃eα|z|2d(z,Γ). Then setting Gk(z)= G(z)/(z−γk), we have sup k Gkα,p <∞ and Gk(γk)eα|γk|2. However, Γ is not interpolating because D+(Γ) = (2α)/π, as follows from the construction of Γ. Interpolation in holomorphic spaces 285 To prove Theorem 1, we will show that Γ is uniformly discrete and that W(Γ) contains only non-uniqueness sequences for Fp α. The result will then follow from an application of Theorem 2. Throughout the proof, we will let Gkdenote an extremal function of the left-hand side of (1). If f∈Fp αvanishes at some point w, we find that f(z)/(z−w)∈Fp α with norm controlled by fα,p, and so e−α|z|2|f(z)||z−w|fα,p. We set f=Gk,z=γkand w=γn, to see that this inequality becomes 1|γk−γn|, whence Γ is uniformly discrete. It remains to prove that W(Γ) contains only non-uniqueness sequences. To begin with, we note that the translation invariance of Fp α along with a normal family argument implies that if Γ meets (1), then so does every member of W(Γ). Thus it suffices to demonstrate that every sequence Γ satisfying (1) is a non-uniqueness sequence for Fp α. Suppose then, that Γ is a uniqueness sequence. If D+(Γ) <(2α)/π, then Γ will be an interpolation sequence and hence a non-uniqueness sequence, so we will assume that D+(Γ) ≥(2α)/π and obtain a contradiction. Fix an arbitrary kand consider the unique function gk∈Fp αwith gk(γn)=eα|γk|2,n=k 0,n=k. Setting g(z)=(z−γk)gk(z), we observe that g(z)/(z−γn)∈Fp αfor arbitrary n. Hence we must have C g(z) z−γn p e−pα|z|2dσ(z)|g(γn)|pe−pα|γn|2.(6) By the subharmonicity of |g(z)/(z−γn)|p, we also have |g(γn)|pe−pα|γn|2)−2<|z−γn|<2 |g(z)|pe−pα|z|2dσ(z) ≤)−2D(γn,2) |g(z)|pe−pα|z|2dσ(z), (7) independently of )>0. For the arguments to be used next, it is convenient to apply a theorem of Landau [4], which implies that we may write instead D+(Γ) = lim sup r→∞ max z∈C m(z,r) r2, 292 A. P. Schuster, K. Seip [2] A. Beurling,“The Collected Works of Arne Beurling”, Vol. 2, Birkh¨auser, Boston, 1989. [3] P. Koosis,“Introduction to HpSpaces”, Cambridge University Press, Cambridge, 1998. [4] H. J. 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Alexander P. Schuster: Department of Mathematics San Francisco State University San Francisco, California 94132 U.S.A. E-mail address:[email protected] Kristian Seip: Department of Mathematical Sciences Norwegian University of Science and Technology N-7491 Trondheim Norway E-mail address:[email protected] Primera versi´o rebuda el 30 d’abril de 1999, darrera versi´o rebuda el 21 de setembre de 1999.