Interpolation and Sampling Hypersurfaces for the Bargmann-Fock space in higher dimensions

We study those smooth complex hypersurfaces W in C^n having the property that all holomorphic functions of finite weighted L^p norm on W extend to entire functions with finite weighted L^p norm. Such hypersurfaces are called interpolation hypersurfaces. We also examine the dual problem of finding all sampling hypersurfaces, i.e., smooth hypersurfaces W in C^n such that any entire function with finite weighted L^p norm is stably determined by its restriction to W. We provide sufficient geometric conditions on the hypersurface to be an interpolation and sampling hypersurface. The geometric conditions that imply the extension property and the restriction property are given in terms of some directional densities.


INTRODUCTION
Let ω = √ −1∂∂|z| 2 denote the standard Euclidean form in C n . Fix a smooth closed complex hypersurface W ⊂ C n and a plurisubharmonic function ϕ such that for some contants C, C ′ > 0, in the sense of currents. For brevity, such an estimate will sometimes be denoted √ −1∂∂ϕ ≃ ω. (1) We say W is an interpolation hypersurface if for each f ∈ bf p ϕ (W ) there exists F ∈ BF p ϕ (C n ) such that F |W = f . (2) We say W is a sampling hypersurface if there is a constant M = M(p, W ) such that for all F ∈ BF p ϕ (C n ), 1 M C n |F | p e −pϕ ω n ≤ W |F | p e −pϕ ω n−1 ≤ M C n |F | p e −pϕ ω n (1) when p < +∞, or a similar estimate involving suprema in place of integrals when p = ∞.
The goal of this paper is to find geometric sufficient conditions for a uniformly flat hypersurface W to be interpolating or sampling. A key concept is given in the following definition. 1 1 vol(B(z, r)) B(z,r) ∂ 2 log |T | ∂ζ i ∂ζ j ω n (ζ) √ −1dz i ∧ dz j .
Remark. Clearly the definition of Υ W (z, r) is independent of the choice of the function T defining W . Moreover, if Θ W is the current of integration associated to W then Υ W is the average of Θ W in a ball of center z and radius r > 0: where 1 A denotes the characteristic function of a set A and * is convolution. Thus, in particular, the trace of Υ W (z, r) is precisely the average area of W in the ball of radius r and center z.
A useful concept in the study of interpolation and sampling for smooth hypersurfaces is the density of these hypersurfaces. Let ϕ r := 1 B(0,r) * ϕ vol(B(0, r)) .
Definition. The density of W in the ball of radius r and center z is On the other hand, a lower bound for D(W, z, r) tells us only that the largest eigenvalue of the form Υ W (z, r) − √ −1∂∂ϕ r (z) is uniformly positive.
Our main results can be stated as follows.
Theorem 1. Let W be a uniformly flat hypersurface. If D + (W ) < 1 then W is an interpolation hypersurface.
Theorem 2. Let W be a uniformly flat hypersurface. If D − (W ) > 1 then W is a sampling hypersurface.
The hypotheses in Theorems 1 and 2 have a geometric interpretation. For simplicity, consider the classical Fock space, which correponds to ϕ = |z| 2 . Then Υ W (z, r)(v, v) is the average number of intersections of the manifold W with a complex line of direction v in the ball of center z and radius r. Thus D + (W ) < 1 means that in any point z and in any direction v the average number of intersecting points between the manifold and a complex line in the direction of v is smaller than some critical value. On the other hand D − (W ) > 1 means that for any point z there is a direction v (which may depend on the point p) such that in the ball of radius r and center z the average number of intersections between W and the complex line with direction v is bigger than some critical value.
Intuitively speaking, the interpretation of our theorems is that if we want W to be interpolating it must be sparse in all points and all directions, but if we want it to be sampling it must be dense in all points, but only in one direction for any given point.
The interpolation and sampling problems in the generalized Fock space have been studied previously. In dimension one there is a full description given in [BO-95] that corresponds to our Theorem 1 and Theorem 2. In dimension 1 the conditions we require are also necessary. This was proved in . It seems plausible that this is also the case in higher dimensions, but the question of necessity remains open.
In several complex variables, there have been many partial and related results. See for instance ,  or . In these works hypotheses are placed on the function T ∈ O(C n ) defining W = Z(T ) in order that W be interpolating in the sense of Berenstein-Taylor, that is to say, any holomorphic function h defined on W and satisfying a growth condition can be extended to an entire function satisfying similar bounds (perhaps with a different constant). For instance a result can be found in  stating that W is interpolating in this sense if Our results do not involve the defining function T , appealing instead directly to the current of intergration defining W . In this sense our results are more geometric in nature. The organization of the paper is as follows. In Section 2 we define and discuss the notion of uniform flatness. In Section 3 we define a non-positive function that is singular along the variety W . As in , this function is used to modify the weight of the Bargmann-Fock space in order to apply the Hörmander-Bombieri-Skoda technique in the proof of Theorem 1. A central point is the use of the Newton potential in the construction. In Section 4 we prove Theorem 1. We begin with the L 2 case. Our approach is to first extend the candidate function to small neighborhoods, and then to patch together these local extensions using the solution of a Cousin I problem with L p bounds. To pass to L p we use results of Berndtsson on L p bounds for minimal L 2 solutions of ∂. A second proof is possible when p = 2, using the method of the Ohsawa-Takegoshi extension theorem. This proof is only mentioned and briefly sketched here. For the details of this approach in the case of the Bergman ball, the reader is referred to . In Section 5 we prove Theorem 2. Finally, in Section 6 we give a simple application of our results to improve on known sufficient conditions for sequences to be interpolating or sampling in C n , n ≥ 2.
Acknowledgement. The authors would like to thank Tamas Forgacs, Jeff McNeal and Yum-Tong Siu for stimulating and useful discussions. Some of this work was done while the first author was visiting the University of Wisconsin, the second author was visiting the University of Michigan, and the third author was visiting Harvard University and the University of Michigan. The authors wish to thank these institutions for their generous hospitality.

UNIFORM FLATNESS
We shall be interested in smooth hypersurfaces W satisfying the following assumption. 3 (F1) There is a positive constant ε o such that the Euclidean neighborhood

Definition.
A smooth hypersurface W satisfying (F1) is said to be uniformly flat.
If we want to extend functions to C n , uniform flatness seems a reasonable condition; we don't want points that are very far apart in W to be very close to each other in the ambient space. When n = 1, W is a discrete set, which is uniformly flat if and only if it is uniformly separated.
For each z ∈ W , denote by T W,z the tangent space to W at z and by n z a unit normal to T W,z in the Euclidean metric ω. Note that n z is determined uniquely up to a unimodular constant. Write for the product of the ε-ball centered at the origin in T W,z , with the ε-disc centered at the origin of T C n ,z and perpendicular to T W,z . We leave it to the reader to verify the following proposition.
Proposition 2.1. Let W be a uniformly flat hypersurface. Then the following hold.
(A) For each R > 0 there is a constant C R > 0 such that for all z ∈ C n , It is not hard to see that every smooth affine algebraic hypersurface is uniformly flat. There are also many non-algebraic examples.

SINGULARIZATION OF THE WEIGHT
As is now standard in L p interpolation problems in several complex variables, one needs to define a strictly plurisubharmonic weight similar to ϕ with singularities along the divisor W . For the sampling problem, one must smooth out this weight near W , while maintaining global bounds away from W .
Our scheme for singularizing the weight follows the method of [BO-95]: we add to our weight ϕ a function s r , called the singularity, to be defined below.
To obtain good properties of the singularity, one needs to use potential theoretic aspects of the ambient space C n . For our purposes, the Newton potential plays a key role. Recall that the Newton potential is the function where c(n) = 1 π n 2 n (n − 1) , For each ζ ∈ C n , this function is harmonic in C n − {ζ} and has the property that The key feature making our approach possible is that this last identity involves only the trace of √ −1∂∂G. It is this fact precisely that links the fundamental solution of ∆ to holomorphic functions on hypersurfaces.
The singularity. Consider the function Since G(z, ζ) is harmonic in each variable separately when |z − ζ| > 0, one sees immediately that Γ is supported on the neighborhood |z − ζ| ≤ r of the diagonal in C n × C n . We define the singularity By the Lelong-Poincaré identity, we have where and We have the following lemma.
Lemma 3.1. The function s r has the following properties.
(3) The function e −2sr is not locally integrable at any point of W .
Proof. By the sub-mean value property for subharmonic functions, Γ r ≤ 0, from which (1) follows. Next, we verify that there is a constant D r such that for all ζ ∈ B(z, r), For this, it suffices to bound the integral Letting ρ = r − |ζ|, we have On the other hand, which demonstrates the bound for I r .
If we now look at z such that |z − ζ| ≥ ε for all ζ ∈ W , (2) follows from the above bound for I r together with formula (2).
To prove (3), we study the singularity of the function in the neighborhood of a point w o ∈ W . Fix Euclidean coordinates t, x 1 , . . . , x n−1 at w o , such that w o is the origin of these coordinates, dt = 0 defines T W,wo and It follows that there are local coordinates ζ 1 , . . . , ζ n−1 on W near w o that are of the form Moreover, the hypersurface W is cut out by a holomorphic function of the form Thus the singularity of the integral (3) is the same as that of This completes the proof.
The proof of (3) in Lemma 3.1 also follows from the uniform flatness of W and the following formula for s r .
and thus Proof. Let α : [0, ∞) → [0, 1] be a smooth compactly supported function which is identically 1 on [0, 1]. Then for R >> r, we have where the second equality follows from the fact that Γ(·, z) is supported on B(z, r). Integrating by parts and letting R → ∞, we have log |T |ω n , as desired.

INTERPOLATION: THE PROOF OF THEOREM 1
Since we assume that i∂∂ϕ ≃ ω, it follows that |ϕ r − ϕ| ≤ C and therefore the spaces BF p ϕ and BF p ϕr are the same space with equivalent norms. The same happens with bf p ϕ and bf p ϕr . Therefore we may assume without loss of generality that in the definition of the densities and thus in the hypothesis of the theorems we have replaced ϕ r by ϕ.

The Cousin I approach.
Local extension. Let ϕ be a plurisubharmonic function in C n with √ −1∂∂ϕ ≤ Mω for some M > 0. Let Ω be a bounded domain in C n containing the origin and denote by H the hyperplane z n = 0. Define where P H denotes orthogonal projection onto H.
provided that the right hand side is finite.
(When p = ∞, the integrals should be replaced by suprema and C p by C.) where R = diam(Ω). In B(0, R) there is a function u such that (1) u is bounded in B(0, R) by a constant depending only on M and R, and .
The result follows.
As a corollary, we have the following lemma.
If p = ∞, then the integrals should replaced by suprema.
Local holomorphic functions with good estimates.

Lemma 4.3. Let ϕ be a function in the unit disk D such that
Then there exist a constant C > 0 and a holomorphic function H ∈ O(D) such that H(0) = 0 and Moreover, if ϕ depends on a parameter in such a way that the bound on ∆ϕ is independent of the parameter, then H can be taken to depend on this parameter in such a way that C does not.
The proof of this lemma, by now well known, can be found in . 8 Construction of the interpolating function. We fix f ∈ bf p ϕ (W ) and ε < ε 0 /2, where ε 0 is as in 2.1-(B). Take a sequence of distinct points {w j ; j = 1, 2, . . .} ⊂ W such that and each point of N ε (W ) is contained in at most a fixed, finite number of the sets B(w j , 2ε).
(We say that the cover is uniformly locally finite.) For convenience of notation we write B i = B w i , 3 2 ε . We add to the cover {B i } i≥1 another open set B 0 = C n \ N1 2 ε (W ). Thus {B j ; j ≥ 0} is a uniformly locally finite open cover of C n . Let {φ i } i≥0 be a partition of unity subordinate to the cover {B i }, i.e., 0 ≤ φ i ≤ 1, supp φ i ∈ B i and i φ i ≡ 1. Moreover we can assume that 2ε) given by Lemma 4.2, and set F 0 ≡ 0. Since the covering {B i } is uniformly locally finite, we have where χ i denotes the characteristic function of B i and, as usual, the symbol means that the left hand side is bounded above by a universal constant times the right hand side. We want to patch together the extensions F i and construct a single holomorphic extension F of f whose norm remains under control. In the standard language of several complex variables, we want to solve a Cousin I problem with L p bounds. The setup of the problem is as follows. For any pair of indices i, j ≥ 0 we define a function G ij in B ij := B i ∩ B j by Observe that If we find such functions G i , then the function F defined by We defineG i ∈ C ∞ (B i ) byG i = j φ j G ij . These functions have all the properties we seek, except they are not holomorphic. We shall now correct the functionsG i by adding to each of them a single, globally defined function.
To this end, note that in B ij we have∂G i =∂G j . Thus there is a well defined∂-closed (0, 1)- Lemma 4.4. One has the estimate Proof. Recall that if ψ is a weight function on the unit disk D such that ∆ψ ≤ K, then there is a constant C such that for any f ∈ O(D), Indeed, the inequality is elementary in the case ψ ≡ 0. Since the Laplacian of ψ is bounded, there is by Lemma 4.3 a non-vanishing holomorphic function g, such that |g| ≃ e ψ . Thus we obtain With this one variable fact it is possible to prove that Only the first inequality is non-trivial. To see how it follows, let T be any entire function that vanishes precisely on W such that dT does not vanish on W . Then by Proposition 3.2, It follows by the density hypothesis that √ −1∂∂ψ r ≃ Id.
Since the function G ij /T is holomorphic in B i ∩ B j , we may apply the one dimensional result above. Let U = B ij ∩ W . Then B ij ≃ U × D(0, ε). We integrate along the slices and apply the one-dimensional result in each disk.
By the density hypothesis, one has the inequality √ −1∂∂(ϕ + s r ) ≥ cω > 0. We will deal first with the case p = 2. It follows from Hörmander's Theorem that there is a function u such that ∂u = h and C n |u| 2 e −2(ϕ+sr) ω n ≤ C W |f | 2 e −2ϕ ω n−1 .
It follows that the holomorphic functions G i =G i − u have the desired properties.
Next we treat the case p ∈ [1, 2). Let us denote ξ = ϕ + s r . Since h is supported away from the singularity of ξ, a look at the definition of h (in particular, it is constructed from certain holomorphic data and cutoff functions) shows that, since h ∈ L p (e −ξ ), h ∈ L ∞ (e −ξ ). It follows that h ∈ L 2 (e −ξ ). Let u be the function of minimal norm in L 2 (e −ξ ) satisfying∂u = h. Then a theorem of Berndtsson [Ber-97, Ber-01] states that u satisfies , provided the right hand side is finite (which in the case at hand applies for p ∈ [1, 2]), and √ −1∂∂ϕ ≃ ω, √ −1∂∂ξ ≥ cω and s r ≤ 0, as is indeed the case here. (We point out that the constants C p in Berndtsson's Theorem depend only on p and on the upper and lower bounds for √ −1∂∂ϕ.) This gives the right bounds for the solution. Moreover, since he −ξ L 2 < +∞, Hörmander's Theorem and the minimality of u tell us that ue −ξ L 2 < +∞. Thus again u|W ≡ 0. Finally, we come to the case p ∈ (2, ∞]. Here we must be a little more careful. Assume first that h ∈ L 2 (e −ξ ) ∩ L p (e −ξ ). Let u be the function of minimal norm in L 2 (e −ξ ) such that∂u = h. Then again by Berndtsson's Theorem u satisfies . This again gives the desired bounds. Moreover, if he −ξ L 2 is finite then by Hörmander's Theorem and the minimality of u, we have ue −ξ L 2 < +∞. Thus again u|W ≡ 0. This proves the result for h ∈ L 2 (e −ξ ) ∩ L p (e −ξ ). To pass to the general case, instead of approximating h we modify the weight ξ. To this end, take any sequence ε j → 0. Since h is identically zero on a neighborhood of W and he −ξ ∈ L p , we have he −ϕ ∈ L p . Thus once again he −ϕ ∈ L ∞ , and by the support of h we have he −ξ ∈ L p . It follows that for all j, he −ξ−ε j z 2 ∈ L 2 . As before, the solution u j to∂u j = h with minimal norm in L 2 (e −ξ−ε j z 2 ) vanishes on W and, by Berndtsson's Theorem, satisfies where the constants C p are independent of j. It follows that u j → u ∈ L p (e −ϕ ). Thus we can construct holomorphic functions F j that extend f and satisfy the estimates By a normal family argument we can take a subsequence F j converging to F ∈ L p (e −ϕ ). The convergence is unifom over compacts and thus F extends f .

Remarks on the twisted∂ approach.
In this section we outline the ideas behind a proof of Theorem 1 in the case p = 2 using the method of the twisted∂ equation.
The idea behind the twisted∂ approach is to replace the usual∂ complex ψ), where the two operators T and S are defined by If the domain Ω ⊂⊂ C n is smoothly bounded and pseudoconvex, then clever manipulation of the usual Bochner-Kodaira identity can be used to show that for any (0, 1)-form u in the domains of T * and S, a twisted Bochner-Kodaira inequality holds: where a = 1 + log(1 + ε 2 ) − log(e sr + ε 2 ), one can deduce from the twisted Bochner-Kodaira inequality an a priori identity which can be used to solve the equation T h = α with estimates Ω |h| 2 e −ψ ω n ≤ C whenever α is an S-closed (0, 1)-form such that for all u with compact support in Ω wheref is any holomorphic extension of f ∈ O(W ) to C n and χ is an appropriate cut-off function, produces a holomorphic function One estimates this function and passes to the limit as Ω → C n , using the Cauchy estimates to pass from L 2 convergence to locally uniform convergence. As already mentioned, the details of this approach will not be fully carried out here. For an adaptation in the case of the Bergman ball, see .

SAMPLING
In this section we prove Theorem 2. As in section 1, we replace ϕ by ϕ r := 1 B(0,r) * ϕ vol(B(0, r)) in the definition of the density and thus in the hypothesis of Theorem 2.

Restrictions and the upper sampling inequality.
Proposition 5.1. If W is a uniformly flat hypersurface, then there is a constant C > 0 such that for all F ∈ BF p ϕ (N ε (W )) one has Proof. By our hypotheses, N ε (W ) is foliated by analytic disks, each of which is transverse to W as well as to the boundary of N ε (W ), and meets W at a single point. For a given x ∈ W , we denote by L x the disk passing through x, and by λ x : D → L x the (unique up to precomposition by a rotation) holomorphic parameterization of L x by the unit disk, sending 0 to x. We begin with the following claim: To see this, let p ∈ L x be a point on the boundary of N ε (W ) that is of minimal distance to x. By definition of N ε (W ), the distance from p to x is at least ε. Let ℓ be the complex affine line in C n containing x and p. Then it follows from our choice of p and from the maximum principle that the projection of L x onto ℓ contains the Euclidean disk in ℓ of center x and radius ε. Thus the area of L x is at least 2πε 2 . Making use of the diffeomorphism which is holomorphic in the second variable, we work on the product W × D.
Let H(x, t) be the function, holomorphic in t, given by Lemma 4.3. That is to say, H(x, 0) = 0 and |Re(H(x, t)) − ϕ(x, 0) + ϕ(x, t)| ≤ C for some positive constant C, since we have assumed that √ −1∂∂ϕ is bounded above. We then have Integration over W then yields |F | p e −pϕ ∧ ω n , 13 and the proof is complete.
Corollary 5.2. If W satisfies (F1) then there is a constant M > 1 such that for every F ∈ BF p ϕ (C n ), The proof of Theorem 2.
The proof will be an almost immediate application of the following sequence of definitions and lemmas.

Definition.
A sequence of complex hypersurfaces W n is said to converge weakly to another complex hypersurface W if the corresponding currents of integration Θ Wn converge to Θ W in the sense of currents.
Lemma 5.3. If W is a uniformly flat complex hypersurface, then for any sequence of translations τ n , the sequence W n = τ n (W ) has a subsequence converging weakly to a uniformly flat complex hypersurface V . Moreover, V has a tubular neighborhood of at least the same thickness as that of W .
Proof. We denote by |Θ Wn | the trace of the current Θ Wn . This is a positive measure that dominates all the coefficients of Θ Wn . By the uniform flatness of W it is clear that for any ball B, sup n |Θ Wn |(B) < C for some constant C depending only on the radius of B. A standard compactness argument produces a subsequence that converges to a positive closed current θ. It remains to show that the limit current θ is a current of integration on a manifold V . This is proved in [B-64], again under the assumptions that for any fixed ball B the mass |Θ Wn |(B) is bounded. Moreover, in this situation the support of Θ Wn converges to V and in any ball the tubular neighborhoods of the W n ∩ B converge to a tubular neighborhood of V ∩ B.

Definition.
A sequence of plurisubharmonic functions ϕ n is said to converge weakly to a plurisubharmonic function ϕ if the corresponding currents √ −1∂∂ϕ n converge to √ −1∂∂ϕ in the sense of currents.
Definition. Given a pair (W, ϕ) where W is a uniformly flat complex hypersurface and ϕ ∈ P SH(C n ) with √ −1∂∂ϕ ≃ ω, we denote by K * (W, ϕ) the collection of all pairs (V, ψ) for which there is a sequence of translations τ n such that τ n (W ) converge weakly to V and ϕ • τ n converge weakly to ψ.
Lemma 5.5. If the pair (W, ϕ) satisfies D − ϕ (W ) = α then all pairs (V, ψ) ∈ K * (W, ϕ) satisfy D − ψ (V ) ≥ α Proof. By hypothesis, for any z ∈ C n and ε > 0 there exists r > 0 and v ∈ C n of unit norm such that We fix an arbitrary z ∈ C n . Take a sequence of translations τ n such that W n = τ n (W ) and ϕ n = τ * n ϕ converge to V and ψ respectively. By definition of D − ϕ (W ) = α, for any ε > 0, there is an r > 0 and unit vectors v n such that By compactness there is a subsequence of the v n converging to v with ||v|| = 1. By Hurwitz's theorem Definition. The pair (V, ψ) is said to be determining if for any f ∈ BF ∞ ψ (C n ), f | V = 0 implies that f ≡ 0.
Lemma 5.6 was essentially proved by Beurling in [Be-89, pp. 341-365], so we omit the proof. This is a key result because it allows us to determine that W is sampling simply by checking the more easily verified condition that V is determining.
Lemma 5.7. If D − ψ (V ) > 1 then the pair (V, ψ) is determining. Proof. Without loss of generality we assume that 0 / ∈ V . In order to arrive at a contradiction, assume there exists an F ∈ BF ∞ ϕ with F |V ≡ 0 and F (0) = 1. By hypothesis there is a direction v such that the density of V in the direction of v is greater than 1. We will work on the line ℓ = Cv. Write f = F |ℓ and φ = ϕ|ℓ, and let Γ = V ∩ ℓ. Then Γ is a uniformly separated sequence with density > 1 with respect to the weight φ. Recall that the one-dimensional lower density is lim inf Since log |f (Re Proof. Denote by BF ∞,0 ϕ+ε|z| 2 the closed subspace of BF ∞ ϕ+ε|z| 2 consisting of functions f such that lim z→∞ |f |e −ϕ−ε|z| 2 = 0.
Lemma 5.10. Let W be a uniformly flat hypersurface. Let Σ be a uniformly separated sequence contained in W . If Σ is a sampling sequence for BF p ϕ then W is a sampling hypersurface for BF p ϕ . Proof. We only need to prove that for any z ∈ W , the inequality holds, where D z = W ∩ B(z, ε), and the constant C may depend on the radius ε of the ball but not on the center z. For if (6) holds then for any function f ∈ BF p ϕ , In order to prove (6) we need the hypothesis that i∂∂ϕ ≃ ω. Under this hypothesis we may again invoke the existence of a non vanishing function h ∈ O(B(z, ε)) such that e ϕ ≃ |h| in B(z, ε) with constants independent of z. Thus, we may replace e −ϕ by h −1 in (6) and get the result if we prove that If D z is a hyperplane then the latter estimate holds for all holomorphic functions g by the sub-mean value property. In a general uniformly flat hypersurface the estimate holds because the distortion introduced in ω n−1 upon rectifying D z by a change of variables is uniformly bounded due to property (B) in Lemma 2.1 for uniformly flat hypersurfaces.
Finally, we are ready to prove Theorem 2. To this end, let ε > 0 be such that D − ϕ (W ) > 1 + ε. We start by proving that W is a sampling manifold for BF ∞ ϕε , where ϕ ε = ϕ + ε|z| 2 . In order to do so, we use Lemma 5.6. We need to check that for any pair (V, ψ) ∈ K * (W, ϕ ε ) the pair (V, ψ) is determining. This is true in view of Lemmas 5.5 and 5.7. Now we take the sequence Σ ⊂ W given by Lemma 5.8. This sequence Σ is a sampling sequence for BF ∞ ϕε and thus it is also sampling for BF p ϕ by Lemma 5.9. Finally by Lemma 5.10 we conclude that W is a sampling manifold for BF p ϕ .

AN APPLICATION TO SEQUENCES IN HIGHER DIMENSIONS
Let ϕ be a plurisubharmonic function in C n such that for some c > 0 Let Γ be a uniformly separated sequence of points in C n . We consider the space Recall that Γ is an interpolation sequence if for each {a γ } ∈ ℓ p ϕ (Γ) there exists F ∈ BF p ϕ (C n ) such that F (γ) = a γ , γ ∈ Γ, and that Γ is a sampling sequence if there is a constant M > 1 such that for all F ∈ BF p ϕ (C n ) 1 M C n |F | p e −pϕ ω n ≤ Γ |F (γ)| p e −pϕ(γ) ≤ M C n |F | p e −pϕ ω n .
Sufficient conditions are known for a sequence to be interpolating, and also sampling. There are also (different) necessary conditions. However, all the known conditions involve only the number of points of the sequence contained in a large ball. It has been known for some time that such a condition could not possibly characterize interpolation and sampling sequences, since it does not take into account how points are distributed relative to one another. For example, consider the situation of interpolation. If all the points of a sequence lie on a line, then to be interpolating there must be at most O(r 2 ) points in any ball of radius r. On the other hand, the number of points of a lattice in C n lying inside a ball of radius r is O(r 2n ). Thus any condition for interpolation that takes into account only the number of points of the sequence lying in a ball of radius r would not suffice to conclude that any lattice, no matter how sparse, is an interpolation sequence. Similar reasoning shows that analogous problems arise in the case of sampling conditions. The present paper and the paper [SV-03] suggest an approach to studying interpolation and sampling sequences by induction on dimension. In [SV-03] two of us tackled the 1-dimensional case. The present paper tackles the problem from the other end. In this section, we show that the results of the present paper already improve what is known for sequences in higher dimension.
6.1. Applications to interpolation. For simplicity, we restrict to the case of sequences in C 2 . As mentioned, at present rather poor density conditions are known in the general higher dimensional case. However, in a very symmetric situation there is a characterization of interpolation and sampling sequences in C 2 . Suppose the sequence Γ is of the form