Multiple Sampling and Interpolation in Weighted Fock Spaces of Entire Functions

We characterize sampling and interpolating sets with derivatives in weighted Fock spaces on the complex plane in terms of their weighted Beurling densities.

where d A denotes the Lebesgue measure on C. We study sampling and interpolation on weighted Fock spaces, where we sample or interpolate using not only the function values, but also the values of its derivatives. Such process is sometimes called multiple or Hermite sampling and interpolation. While sampling and interpolation theory provides a mathematical foundation for the tasks of digitalization and encoding of analog signals into bit-streams, the use of derivatives incorporates the trend of the signal as well, and is well studied in Paley-Wiener spaces and shift-invariant spaces on the real line; see, e.g., [1,2,12,21].
In this article, we characterize the sets that allow for multiple sampling and interpolation in weighted Fock spaces in terms of certain densities, provided that the number of derivatives considered at each sampling point is bounded. The precise assumptions and definitions are given in Sect. 2.
The prime example of a Fock space is the Bargmann-Fock space, where φ(z) = α|z| 2 , and α > 0 [26]. For this weight, the Heisenberg group acts irreducibly on F 2 φ , which results in a important homogeneity property called translation invariance [26,Section 2.6]. More precisely, F 2 φ is invariant under the so-called Bargmann-Fock shifts [26]: (1.1) Necessary and sufficient conditions for sampling and interpolation without derivatives are fully described in terms of planar Beurling densities [23,24]. Specifically, the lower and upper Beurling densities of a set ⊆ C are where B(z, R) denotes the Euclidean ball with center z and radius R. For φ(z) = α|z| 2 and a separated set , the existence of two sampling constants A, B > 0 leading to a sampling inequality is completely characterized by the density condition D − ( ) > α π . Similarly, the density condition D + ( ) < α π completely characterizes the validity of the following interpolation property: given a sequence {c λ : λ ∈ } ⊂ C satisfying the growth condition λ∈ |c λ | 2 e −φ(λ) < ∞ there exists (at least) one function f ∈ F 2 φ such that (1.5) Slightly more technical formulations of these density characterizations apply to arbitrary, possibly non-separated sets [23,24]. These results parallel Beurling's sampling and interpolation theorems for the Paley-Wiener space of square integrable functions on the real line having Fourier transforms supported on the unit interval, with the difference that the latter results do not provide a complete characterization in terms of densities, as the necessary density condition involves a non-strict inequality while the sufficient involves a strict one [6,7]. 1 Translation invariance plays a key role in [23,24], and the arguments do not seem to extend easily to more general weights. For the classical weight φ(z) = α|z| 2 , sampling and interpolation with bounded multiplicities are studied in [9], while unbounded multiplicities are treated in [8]. In this setting the sampling (1.4) and interpolation equations (1.5) involve not only the function f but also its translation covariant derivative Sampling and interpolation on Fock spaces with general weights has been studied in [5] and [19]. The characterization is in terms of variants of the Beurling densities (1.2), (1.3), where the factor π R 2 is replaced by the weighted measure (1.7) As ∂∂ α|z| 2 = α, Beurling densities defined using (1.7) extend the classical ones (1.2), (1.3) up to normalization. For general weights, the lack of translation invariance demands new techniques, such as the introduction of approximate translation operators [19], and∂-surgery together with Riesz decompositions [5] (see Sect. 3).
In this article we simultaneously extend the results from [5,9,19] to study sampling and interpolation with multiplicities on Fock spaces with general weights. Our starting point is the observation that, for the classical weight φ(z) = α|z| 2 , the covariant derivative (1.6) is the formal adjoint of the∂-operator with respect to the scalar product that induces the norm of F 2 φ . Explicitly, We consider this operator for general weights φ, and study sampling and interpolation involving iterated applications of∂ * φ . We derive a characterization of these properties in terms of a suitable variant of the Beurling densities (1.2), (1.3), that counts points multiply according to the number of concerned derivatives as in [9], and weights balls with the Laplacian factor (1.7) as in [5,19].

Motivation
One main motivation for our results is validating the differential operator (1.8) as an adequate replacement for the covariant derivative (1.6), by showing that it leads to similar sampling and interpolation properties. Other candidates for this role are the differential operators defined as ordinary differentiation at the origin, conjugated with the approximate translation operators from [19]. This second choice is however unnatural, as it arbitrarily distinguishes the origin, and leads to complicated compatibility issues (stemming from the fact that approximate translation operators are not associated with a group representation). On the other hand, although they lack the simplicity of (1.8), differential operators associated with approximate translations would allow a more straightforward generalization of the proofs of [5,19]. In any case, one can use our present results to show a posteriori that both differential operators -(1.8) and the ones defined in terms of approximate translations -do lead to similar sampling and interpolation theorems.
Fock spaces find important applications in the simultaneous time-frequency analysis of real variable functions. The most successful applications concern the classical weights and Gabor systems, which are structured functional dictionaries for L 2 (R), generated by the Gaussian function g(t) = e −π t 2 , t ∈ R, and their time-frequency shifts along a given set of nodes ⊆ R 2 . By means of the Bargmann transform, the spanning properties of (1.9) can be reformulated as sampling and interpolation properties of on the Fock space with weight φ(z) = π |z| 2 [10], and thus characterized in terms of Beurling densities [18,23,24]. Multiple sampling and interpolation on F 2 φ concerns in turn the spanning properties of multi-window Gabor systems, where the Gaussian function is supplemented with additional Hermite functions [9, Remark 1]. Other weights φ(z) = απ|z| 2 allow a similar analysis for the general Gaussian function g(t) = e −at 2 . Applications of more general weighted Fock spaces to timefrequency analysis are more recent, and mainly connected to non-linear properties of the so-called short-time Fourier transform, where for example weights are chosen in a signal dependent fashion, see e.g., [13]. We expect our current results to find applications in this direction.

Organization and Technical Overview
In Sect. 2 we present the precise definitions and assumptions as well as the main results. Sect. 3 provides useful estimates from potential theory, which are important to treat derivatives of functions in Fock spaces. The sufficiency of the density conditions for sampling and interpolation is shown under additional technical assumptions in Sect. 4, following closely the arguments from [5]. The most important task here is showing that the operator (1.8) is suitably compatible with that line of argument, which is based on Hörmander's L 2 -estimates for ∂ [16]. The necessity of the density conditions for multiple sampling and interpolation in the weighted case is challenging, as the classical approach relies on translation invariance [9], and the one in [19] on a subtle form of approximate translation invariance. Instead of adapting the delicate proof from [19], in Sect. 5 we develop a perturbation argument that allows us to reduce the problem to the case of no multiplicities. This line of reasoning seems to provide a more direct proof of the necessary density conditions for multiple sampling and interpolation even for the classical weights [9]. The main results are finally proved in Sect. 6, where the remaining technical assumptions are removed. Section A is an appendix listing auxiliary results related to iterated derivatives of composite functions.

Sets with Multiplicity
A set with multiplicities is a pair ( , m ), where ⊆ C, and m : → N is a bounded function called multiplicity function. In problems concerning evaluations of functions at , the number m (λ) indicates how many derivatives are involved at the point z = λ. More precisely, m (λ) = k indicates the interpolation or sampling of a function and its first k − 1 derivatives at λ.

Notation
Complex disks are denoted B(z, r ) := {y ∈ C : |z − y| < r }, with z ∈ C and r > 0. We use the notation x y if there exists a constant such that x ≤ Cy. The constant is normally allowed to depend on sup λ∈ m (λ) and φ, while other dependencies are noted explicitly. The precise value of unspecified constants may vary from line to line.

Separated Sets
and it is called (uniformly) separated if Separated sets are relatively separated, and relatively separated sets are finite unions of separated sets. For simplicity of notation, we write ρ instead of ρ( ) when no confusion can arise. We say that a set with multiplicities ( , m ) is separated (respectively relatively separated) if is separated (respectively relatively separated).

Wirtinger Derivatives and the@ * Operator
We normalize the Laplacian as where ∂ and∂ denote the Wirtinger derivatives: For f ∈ C 1 (C) we define the operator which is the formal adjoint of the∂-operator with respect to the weighted scalar product When the dependence on φ is clear from the context, we write for short∂ * . For the classical weight φ(z) = α|z| 2 ,∂ * φ f is the covariant derivative (1.6).

Interpolating Sets and Sampling Sets
We For the classical weight φ(z) = α|z| 2 , using the covariance property of ∂ * φ , it is easy to see that where e j (z) = α j+1 /π z j and W w denotes the Bargmann-Fock shift by w (1.1). Multiple sampling and interpolating sets with respect to the weight φ(z) = α|z| 2 are defined in [9] in terms of Bargmann-Fock shifts of the normalized monomials f j (z) = (α j / j!) 1/2 z j . Since, as in [9], we only consider sets with bounded multiplicity, for the classical weight φ(z) = α|z| 2 , the present notion of multiple sampling and interpolation is equivalent to the one from [9].

Beurling Densities
Given a set with multiplicities ( , m ), n (z, r , , m ) is defined as the number of points of the set in the disk B(z, r ), counted with multiplicities. That is, is a weighted sum of Dirac deltas, and 1 B(0,r ) is the characteristic function of the disk. The lower and upper Beurling densities of ( , m ) with respect to φ are: (2.2)

Compatibility Assumptions
We always assume that sets with multiplicities have bounded multiplicity functions and denote We say that ( , m ) and φ are compatible if the following conditions hold. For n = 0 or n = 1, we only require that m < φ < M for some positive constants m, M. For n ≥ 2, we additionally require φ ∈ C n +1 (C) and sup z∈C |∂ The conditions are chosen so that there exists ε r > 0 for which for every ζ ∈ C, z ∈ B(ζ, ε r ) and 0 ≤ j ≤ n .

Main Results
Our goal is to derive the following characterization of sampling and interpolating sets.

Riesz Decomposition
The Riesz decomposition, presented below, describes the structure of subharmonic functions on bounded subdomains of the complex plane. It is an essential tool to study weighted Fock spaces, as their very definition involves a subharmonic function.

Theorem 3.1 (Riesz decomposition) Let D be a domain in C, and let u : D → R be a subharmonic function. If K ⊆ D is an open and relatively compact set, then there exists a harmonic function h on K such that
The function G[ u] in (3.1) is called the logarithmic potential of u on the domain K . For more on Riesz decomposition, see [14,20,22].

Lemma 3.2 Let ⊂ R n be open and bounded. Let
Here, As an application, we have the following bounds for Riesz decomposition.
Proof Let z ∈ B(λ, ε). We first note that Note also that Consequently, We now show similar bounds for the derivatives of the logarithmic potential of φ. Suppose n ≥ 1. As shown in [11,Lemma 4 Suppose now that n ≥ 2. Recall that in this case, φ is assumed to be C n +1 , thus the Laplacian is C n −1 . By Theorem 3.1, and that there exists a constantC 2 > 0 for which where the right-hand of the last inequality is uniformly bounded as a consequence of (3.8) and the assumption that ∂ φ is uniformly bounded. Iterating this argument, we obtain that where the right-hand is uniformly bounded. This proves (3.4). Finally, since G is holomorphic, repeated differentiation of (3.6) yields, Thus (3.4), gives (3.5).
The following lemma helps profit from Riesz decomposition in estimates involving derivatives. Then there exist a constant C = C φ,K , such that for every λ ∈ ∩ K , and 0 ≤ j ≤ n , the following estimates hold: Proof By Leibniz rule: This yields (i); (ii) follows similarly.
Finally, we note that Cauchy bounds extend to weighted derivatives.

Lemma 3.5 Let ( , m ) be a separated set with multiplicities that is compatible with φ.
Then for each λ ∈ and j ≤ n , where the decomposition is taken on the set B(λ, 1). We apply Lemma 3.4, together with Cauchy's bound to obtain

Sufficient Conditions for Interpolation and Sampling
Throughout this section we assume that ( , m ) is a separated set with multiplicities that is compatible with φ.

Interpolation
We start by showing that interpolation can be solved locally.
Proof We consider G λ as in the Lemma 3.3 on the set B(λ, ε). We define and k j are real coefficients to be chosen so that To obtain an explicit formula of k j we proceed as follows, The n-th derivative of two composite functions can be computed by means of Faà di Bruno's formula and the Bell polynomials (see Eq. (A.6) and Section A for definitions and suitable references). More precisely, where B k is the k−th complete Bell polynomial. We write B g k (w) := B k ∂ g(w), . . . , ∂ k g(w) to shorten notation.
On account of the above remark we conclude that By Lemma 3.3, we finally obtain For technical reasons we now study the interpolation problem on F 2 φ with respect to the operator∂ * φ associated with a second weightφ. Provided that both weights are sufficiently smooth, powers of both operators are formally related bȳ (4.8) ( , m ) be a separated set with multiplicities that is compatible with φ. Assume additionally that φ is continuous. Let χ r = 1 πr 2 1 B(0,r ) and ν := λ∈ m (λ) · δ λ . Suppose that there exists δ > 0 and r > 0 such that
Let ψ := φ + πv. The modified weight ψ has the properties and for each λ ∈ the following inequality holds when z ∈ B(λ, ρ/2), for some constant C ρ that depends on ρ( ) and r , although this second dependency is not stressed in the notation. (Notice the factor m (λ) in front of the logarithm.) The first step is to construct the non-analytic interpolant. Fix λ ∈ . Note first that, by (4.10) and (4.11), ( , m ) is compatible withφ. By Lemma 4.1 applied to the weightφ, there is an analytic function f λ : and, due to (4.10), for all z ∈ B(λ, ρ/2) where the constant C ρ is independent of λ. Now we patch these functions together. We let g ∈ C ∞ 0 be 1 on B(0, ρ/4) and 0 outside B(0, ρ/2), satisfying |∂ g| < C ρ . Then solves the interpolation problem, although this function might not be analytic.
We consider the problem ∂u = ∂ f . By Hörmander's estimate, there exists a solution satisfying For the first inequality we used that ψ ≤ φ. For the second we used that f is the solution provided by Hörmander's estimate 2 and that ψ > δ. For the third, we used (4.12) and the fact that ∂ f is zero on a ρ/4 -neighborhood of . The last inequality follows from the separation of and (4.13).

Sampling
Since φ is bounded, we can choose t so close to 1 that and for E ν χ r we obtain a similar expression. Using that log |z − w| 2 is harmonic on w ∈ B(λ, ε) when d(z, λ) > r + ε and that the set is relatively separated, we obtain that |v| ≤ C ε . Moreover, ε). Notice the factor m (λ) in front of the logarithm. Let h ∈ F 2 φ . As shown in [5, Eq. 3], we have Thus, Let g λ = he −G λ , where G λ is as in Lemma 3.3, with respect to the set B(λ, 1). Then We now use the (m (λ) − 1)-th order Taylor expansion of g λ on B(λ, ε) (4.14) We estimate the m (λ)-th term by Cauchy estimate The remaining terms in (4.14) are estimated observing that for any w ∈ B(λ, ε) where B j is the j−th Bell polynomial. Evaluating at λ we obtain: Thus, for each j ≤ m (λ) − 1, and applying Remark 4.1 we have Putting everything together, we obtain The relative separateness of implies Therefore, Choosing ε small enough, the last term can be absorbed into the left-hand side, yielding the desired sampling estimate.

Necessary Conditions for Interpolation and Sampling
In this section we reduce the problem of deriving necessary conditions for interpolation and sampling with derivatives to the corresponding problem without derivatives. We assume throughout this section that ( , m ) is a separated set with multiplicities that is compatible with the weight φ.
Our arguments are based on inspecting Taylor expansions. The following observation will be used repeatedly. ε ∈ (0, 1/4), λ ∈ C and F : U ⊆ C −→ C be holomorphic, where B(λ, 1) ⊆ U . The Taylor expansion of F of degree n at λ,

Remark 5.1 Let
In particular, if λ is such that |λ − λ | = ε and w ∈ C is such that As a first step towards the necessary conditions we compute derivatives of weighted functions.

Interpolation
We now show that a weighted interpolating set can be modified to reduce its maximal multiplicity while preserving its density and interpolating property. ( , m ) be a separated set with multiplicities that is compatible with φ. Suppose that ( , m ) is interpolating for F 2 φ (C), and that sup λ∈ m (λ) = n + 1 ≥ 2. Then there exists a separated and interpolating set (˜ , m˜ ) such that sup λ∈˜ m˜ = n and D ± ( , m ) = D ± (˜ , m˜ ).

Proof
Step 1. (Definition of the new set).
Since ε < ρ( )/2 it follows easily that˜ is separated. It is also clear that sup m˜ = n .
For each λ ∈ max we let b λ be defined by The sequence satisfies We define a new sequence a = a (λ, j) λ∈ , 0≤ j≤m (λ)−1 ⊆ C by: otherwise.
By (5.5), Since ( , m ) is an interpolating set, there exists a function f ∈ F 2 φ such that The first two conditions, together with (5.6) yield (Q.1) and (Q.2). To check (Q.3), we let λ ∈ max , and note that, in terms of f , (5.4) reads:

Applying (5.3) to the last equation we obtain
We apply Remark 5.1 to F = f e −H λ and conclude from (5. 2) that . Therefore, which yields (Q.3).
Step 3. We show that ˜ , m˜ is an interpolating set.
if λ ∈ and j = 0. The constructed sequences satisfy and, m˜ ) .
To see that interpolates with the right values on , we observe that for each λ ∈ max , (λ ,0) , when λ ∈ max . For λ ∈ \ we argue as follows. Since We have thus shown that (˜ , m˜ ) is an interpolating set.
The number of points of the set in the disk of center z and radius r , counted with multiplicities, satisfies n (z, r , , m ) ≤ n z, r + ε,˜ , m˜ . Since 0 < m ≤ φ ≤ M then for any δ > 0, there exists r δ such that if r > r δ , then We let δ → 0. Exchanging the roles of and˜ we also see that

Sampling
The modification of sampling sets to reduce their multiplicity is based on the following perturbation lemma.
We can now modify a sampling set to reduce its maximal multiplicity without altering its sampling property. ( , m ) be a separated set with multiplicities that is compatible with φ. Assume that ( , m ) is sampling for F 2 φ (C), and that sup λ∈ m (λ) = n + 1 ≥ 2. Then there exists a separated set with multiplicities (˜ , m˜ ) that is sampling for F 2 φ , and such that sup λ∈˜ m˜ (λ) = n and D ± ( , m ) = D ± (˜ , m˜ ).

Proposition 5.4 Let
Proof Since ( , m ) is a sampling set for F 2 φ (C), Let 0 < ε < min{ρ( )/2, 1/4} and consider max , ,˜ and m˜ as in the proof of Proposition 5.2. Hence, D ± ( , m ) = D ± (˜ , m˜ ), and˜ is separated. As a consequence, (˜ , m˜ ) satisfies an upper sampling bound. We now show that if ε is small enough, a lower sampling bound also holds for (˜ , m˜ ).

Proof of the Main Results
Proof of Theorem A Suppose that ( , m ) is set of interpolation for F 2 φ . Then the unweighted set is also a set of interpolation for F 2 φ , and therefore separated; see, e.g., [19,Prop. 3]. Repeated application of Proposition 5.2 shows that there exists a set of interpolation˜ ⊆ C without multiplicity such that D + φ ( , m ) = D + φ (˜ ). As shown in [19,Theorem 2],˜ must satisfy D + φ (˜ ) < 1/π . Note that in [19], the space F 2 φ is denoted F 2 2φ . Towards the sufficiency of the density conditions, let ( , m ) be a separated set with multiplicities, satisfying D + φ ( , m ) < 1/π . Hence, for some ε > 0 we have D + φ ( , m ) < 1/π − ε. Thus, for a certain R > 0 and for all z ∈ C, where the bar in the integral denotes an average (division by the total measure). Let χ R = π R 2 −1 1 B(0,R) ,φ = φ χ R , and ν := λ∈ m (λ) · δ λ . Then We wish to apply Proposition 4.2, with the roles of the weights φ andφ interchanged. The density condition (4.9) is verified forφ, because of the previous equation. In addition, φ is continuous, and φ is bounded above and below with the same constants that bound φ. Moreover, we claim that To see this, let z 0 ∈ C and let be the Riesz decomposition of φ on B(z 0 , R + 1), as given by Theorem 3.1. Let z ∈ B(z 0 , 1). We average (6.2) over B(z, R) and use the mean value property for harmonic functions to concludẽ and, therefore, The desired number of derivatives of each term on the right-hand side of the previous equation is bounded independently of z 0 by Lemma 3.3, (3.4) (the bound does depend on R). Hence, (6.1) follows. We can therefore apply Proposition 4.2, with the roles of the weights φ andφ interchanged, and conclude that ( , m ) is an interpolating set for , and the interpolation property is considered with respect to∂ * φ .)

Proof of Theorem B
Suppose that ( , m ) is sampling for F 2 φ . Then, for some constant The upper sampling bound without multiplicities already implies that is relatively separated; see, e.g., [19,Prop. 1]. The existence of a separated set ⊆ which is also sampling for F 2 φ follows from standard arguments (see for instance [3,Theorem 3.7]).
Without loss of generality we assume that is already separated. By repeated application of Proposition 5.4, there is a sampling set˜ ⊆ C without multiplicities, such that D − φ ( , m ) = D − φ (˜ ). As shown in [19,Theorem 1],˜ must satisfy D − φ (˜ ) > 1/π . Conversely, suppose that ( , m ) is a relatively separated set with multiplicities containing a separated subset , satisfying D − φ , m > 1/π . The upper sampling bound follows from the fact that is relatively separated and Lemma 3.5. For the lower sampling bound we assume without loss of generality that = . Proceeding as in the proof of Theorem A, we considerφ = φ χ R , for some large R > 0 and conclude from Proposition 4.3, that ( , m ) is a sampling set for F 2 φ . By (6.1), F 2 φ and F 2 φ contain the same functions and have equivalent norms. Moreover, for each f ∈ F 2 φ , we combine (4.8) and (6.1) to conclude that This gives the desired bound.
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If n ≥ 1, the sum In what follows, we use the abbreviation