Marcinkiewicz-Zygmund inequalities for polynomials in Fock space

We study the relation between Marcinkiewicz-Zygmund families for polynomials in a weighted L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-space and sampling theorems for entire functions in the Fock space and the dual relation between uniform interpolating families for polynomials and interpolating sequences. As a consequence we obtain a description of signal subspaces spanned by Hermite functions by means of Gabor frames.


Introduction
We study sampling and interpolation in Fock space and the relation to sampling and interpolation of polynomials. The Fock space F 2 consists of all entire functions with finite norm where dm(z) = dxdy is the Lebesgue measure on C ≃ R 2 . We denote by P n the holomorphic polynomials of degree at most n. A sequence of (finite) subsets Λ n ⊆ C is called a Marcinkiewicz-Zygmund family for the P n in Fock space F 2 , if there exist constants A, B > 0, such that for all n large, n ≥ n 0 , for all p ∈ P n .
Here k n is the reproducing kernel of P n , when endowed with the inner product inherited from F 2 . This notion corresponds to the standard definition of sampling in a reproducing kernel Hilbert space H. Let k λ (z) = k(z, λ) be the reproducing kernel of H, i.e., f (λ) = f, k λ H at the point λ. Then a sequence Λ is a sampling set for H, if the normalized reproducing kernels k(z,λ) In the Fock space F 2 the reproducing kernel is k(z, w) = e πzw , and a sequence Λ ⊆ C is sampling in F 2 , if and only if In this article we compare the notion of Marcinkiewicz-Zygmund families for P n with sampling sequences for the Fock space F 2 . We will see that both notions are intimately connected. Roughly speaking, suitable finite sections of a sampling set for F 2 yield a Marcinkiewicz-Zygmund family for the polynomials P n in F 2 , and suitable limits of a Marcinkiewicz-Zygmund family yield a sampling set for F 2 .
A precise formulation is contained in our main result. (See Section 5 for an explanation of weak limits.) Theorem 1.1. (i) Assume that Λ ⊆ C is a sampling set for F 2 . For τ > 0 set ρ n such that πρ 2 n = n + √ nτ and let B ρn be the centered disk of radius ρ n . Then for τ > 0 large enough, the sets Λ n = Λ ∩ B ρn form a Marcinkiewicz-Zygmund family for P n in F 2 .
(ii) Conversely, every weak limit of a Marcinkiewicz-Zygmund family (Λ n ) for P n in F 2 is a sampling set for F 2 .
A dual result establishes a similar relationship between interpolating sets for F 2 and uniformly interpolating sets for P n . A set Λ is interpolating in F 2 , if for every sequence (a λ ) λ∈Λ ∈ ℓ 2 (Λ), there exists f ∈ F 2 such that f (λ)e −π|λ| 2 /2 = a λ for all λ ∈ Λ. Of course, for polynomials of degree n every set of n + 1 points is interpolating. In analogy to the definition of Marcinkiewicz-Zygmund families we call a family of (finite) subsets Λ n ⊆ C a uniform interpolating family, if there exists a constant A > 0, such that for every sequence a = (a λ ) λ∈Λn ∈ ℓ 2 (Λ n ) there exists a polynomial p ∈ P n such that p(λ)k n (λ, λ) −1/2 = a λ for λ ∈ Λ n with norm control p 2 F 2 ≤ A a 2 2 . Theorem 1.2. (i) Assume that Λ ⊆ C is a set of interpolation for F 2 . For τ > 0 and ǫ > 0 set ρ n , such that πρ 2 n = n − √ n( √ 2 log n + τ ). Then for every τ > 0 large enough, the sets Λ n = Λ ∩ B ρn form a uniform interpolating family for P n in F 2 .
(ii) Conversely, every weak limit of a uniform interpolating family (Λ n ) is a set of interpolation for F 2 .
Marcinkiewicz-Zygmund families and uniform interpolating families arise several areas of analysis. They can be understood as finite-dimensional approximations of sampling theorems in reproducing kernel Hilbert spaces. Theorem 1.1(ii) shows that a Marcinkiewicz-Zygmund family can be used to prove a sampling theorem in an infinite dimensional space. In approximation theory, a Marcinkiewicz-Zygmund family for a sequence of nested subspaces gives rise to a sequence of quadrature rules and function approximation from point evaluations, see [11] for this aspect. Random constructions of Marcinkiewicz-Zygmund families occur in the theory of deterministic point processes, e.g., [1,2,4,5]. In the recent advances in complexity theory and data analysis Marcinkiewicz-Zygmund families are implicit in the discretization of norms. For a nice survey see [15]. This work has several predecessors in different contexts. In [12] we have studied the analogous problem in the Bergman space and in the Hardy space in the unit disk. Indeed, our proof strategy for Theorem 1.1 is taken from [12]. Whereas in Bergman space the results can be formulated similarly to Theorem 1.1, the situation is Hardy space is rather different and the construction of Marcinkiewicz-Zygmund families needed to be based on different principles. In [10] a sampling theorem for bandlimited functions was derived via Marcinkiewicz-Zygmund families for trigonometric polynomials. The set-up of [16] is a compact manifold with a positive line bundle. Marcinkiewicz-Zygmund families for the space of holomorphic sections in powers of the line bundle are connected to sampling sequences in the tangent space.
Though line bundles appear much more complicated objects than Fock space, which even has a closed-form reproducing kernel, Fock space presents some new difficulties. It lacks compactness that made off-diagonal estimates for the reproducing kernel easier in [16]. Another source of difficulty is the behavior under translation. The Fock space F 2 is invariant with respect to Bargmann-Fock shifts, while P n endowed with the Fock norm is not.
Finally we mention the extensive work on the asymptotics of reproducing kernels for weighted polynomials and related construction of random Marcinkiewicz-Zygmund families in the context of determinental point processes [1] - [5]. In [3,4] Y. Ameur and his coauthors have studied a similar notion of sampling polynomials with respect to the discrete norm λ∈Λn |p(λ)| 2 e −πn|λ| 2 instead of λ∈Λn |p(λ)| 2 kn(λ,λ) . Note that all this work uses measures that depend on the polynomial degree n, quite in contrast to our set-up (2). Their choice was motivated by problems arising in random Gaussian matrix ensembles and models of the distribution of points in the one component plasma. The common ground is the construction of point sets that are sampling for polynomials in Fock spaces. In [3,4] this is achieved with random processes (determinental point processes), whereas our construction is purely deterministic and investigates the connection to the infinite-dimensional sampling problem in F 2 . Therefore the results in [3,4] are not directly comparable to ours.
One of our basic tools is the size of the reproducing kernel for the polynomials in a weighted L 2 -space. Since our weight is the Gaussian weight, the kernel can be expressed explicitly in terms of the incomplete Gamma function which is a classical and well-studied object. We have collected the necessary results in the appendix for the sake of being self-contained. Estimates for the reproducing kernel have been studied in great generality in [1,5] with potential theoretic methods -without any reference to the incomplete Gamma function. Possibly these estimates could also be used in our context.
The estimates for the intrinsic reproducing kernel k n show that (i) the L 2 -energy of a polynomial of degree n is concentrated in a disk of radius n/π, the socalled bulk region, and (ii) that the intrinsic kernel k n for P n is comparable to the kernel k(z, w) = e πzw precisely in the bulk region. See Lemma Lemma 2.2 and Corollary 3.2 for the precise statements.
As a consequence of Theorem 1.1 we mention an application to time-frequency analysis. It is well-known that all problems about sampling in Fock space possess an equivalent formulation about Gabor frames in L 2 (R). To state this version, we denote the time-frequency shift of a function g by z = (x, ξ) ∈ R 2 by (π(z)g)(t) = e 2πiξt g(t − x) for t, x, ξ ∈ R. The L 2 -normalized Hermite functions are denoted by h n , in particular h 0 (t) = 2 1/4 e −πt 2 is the Gaussian. Then Theorem 1.1(i) is equivalent to the following statement, which may be of interest in the time-frequency analysis of signal subspaces [14]. Theorem 1.3. Assume that Λ is a sampling set for F 2 and τ > 0 large enough. Then {π(λ)h 0 : π|λ| 2 ≤ n + √ nτ } is a frame for V n = span {h k : k = 0, . . . , n} with bounds independent of n. This means that Outlook. It is needless to say that the topic of Marcinkiewicz-Zygmund families and sampling theorems admits dozens of variations. The ultimate goal is to understand Marcinkiewicz-Zygmund families for polynomials P n in a weighted Bergman space on some general domain X ⊆ R d (or ⊆ C d ). Intermediate problems would be Marcinkiewicz-Zygmund families for polynomials in Fock spaces with more general weight e −Q(z) , or the construction Marcinkiewicz-Zygmund families for multivariate Bergman spaces A 2 (B n ) in n complex variables on unit ball in C n . Even simple variations of the set-up yield interesting new questions.
The paper is organized as follows: In Section 2 we recall the basic facts about the Fock space and the associated reproducing kernels. Section 3 summarizes the required asymptotics of the incomplete Gamma function. In Section 4 we relate sampling sets for Fock space to Marcinkiewicz-Zygmund families and prove the first part of Theorem 1.1. Section 5 covers the converse statement. In Section 6 we deal with uniform interpolating families and prove Theorem 1.2. The connection to the time-frequency analysis of signal subspaces is explained in Section 7. Finally, in the appendix we offer some elementary estimates for the zero-order asymptotics of the incomplete Gamma function. These are, of course, well-known and added only to make the paper self-contained.

Fock space
The monomials z → z k are orthogonal in F 2 , and the normalized monomials form an orthonormal basis for F 2 . Let P n be the subspace of polynomials of degree at most n in F 2 . The reproducing kernel of P n is given by As n → ∞, this kernel converges to the reproducing kernel of F 2 : As we have learned in our study of Marcinkiewicz-Zygmund families in Bergman spaces [12], we will need to understand the relation of the kernel k n to k. For this purpose we will make use of the properties and the asymptotics of the incomplete Gamma function Denote the centered disc of radius ρ by B ρ = {z ∈ C : |z| ≤ ρ}. Then Proof. See [21, 8.4.8], or use the obvious formula 1 n! Γ(n + 1, repeatedly and then use analytic extension and substitute r = πzw. The energy of a polynomial p(z) = n k=0 a k z k ∈ P n on a disc B ρ is 1 Note that we consider P n as a subspace of F 2 and always use the fixed weight e −π|z| 2 . The work on determinental point processes always uses the weight e −πn|z| 2 for P n . The formulas and the asymptotics are therefore different.
In the last inequality we have used the fact that k → γ(k+1,πρ 2 ) k! is decreasing and that z k 2 F 2 = k!/π k by (6). Consequently, the energy of a polynomial p ∈ P n is concentrated B ρ with ρ = n/π, the so-called bulk region.
Lemma 2.2. For every p ∈ P n we have Proof. This follows immediately from the previous estimates via

Asymptotics of the incomplete Gamma function
The asymptotic behavior of the incomplete Gamma function is well understood. We collect the properties required for Marcinkiewicz-Zygmund families in Fock space. As usual f ≍ g means that there exists a constant such that The following result has been proved on several levels of generality [9,19,20,27,28].
The normalized incomplete Gamma function admits the asymptotic expansion We only need these estimates for a = n + 1 and τ > 0, but their validity has been established for large domains in C.
In fact, C(τ ) can be taken as The convergence is uniform on bounded sets ⊆ R + and exponentially fast.
Proof. Items (i) and (ii) follow readily from (10) as follows: (v) is taken from [21], formula 8.10.13. For completeness we summarize the arguments for the zero order asymptotics in the appendix. In contrast to the full asymptotics of the incomplete Gamma function, they are elementary.
depending only on τ , but not on n.
We also need an off-diagonal estimate for the kernel k n .
(i) A set Λ ⊆ C is sampling for F 2 , if and only if it contains a uniformly separated set Λ ′ ⊆ Λ with lower Beurling density D − (Λ ′ ) > 1.
(ii) If Λ is relatively separated, i.e., a finite union of K uniformly discrete subsets of C with separation ρ > 0, then Proof. Lower bound : Since always k n (z, z) ≤ k(z, z) = e π|z| 2 , we may replace k n by k in the sampling inequalities: Since Λ is a sampling set for F 2 , the first term satisfies λ∈Λ |p(λ)| 2 e −π|λ| 2 ≥ A p 2 F 2 . For the second term we observe that Λ is a finite union of uniformly discrete sets with separation ρ > 0 and apply (13) and (7): Our choice of ρ n implies that . In view of Corollary 3.1(i) we may choose τ ′ and hence τ so that λ∈Λ:|λ|>ρn for large n, n ≥ n 0 , say. Combining the inequalities, we obtain λ∈Λn |p(λ)| 2 kn(λ,λ) ≥ A 2 p 2 F 2 for all p ∈ P n . Upper inequality: For the above choice of τ Proposition 3.1(ii) says that because Λ ⊇ Λ n is a sampling set for F 2 .
Note that the lower bound in the Marcinkiewicz-Zygmund inequalities matches the lower bound A of the sampling inequality in F 2 , whereas the upper bound is 4 erfc (τ / √ 2) −1 B depends also on the additional parameter τ .
For n large enough and τ / √ n < δ, we find that For a Marcinkiewicz-Zygmund family for P n we need at least dim P n = n + 1 points in each layer Λ n . The construction above yields Marcinkiewicz-Zygmund families for Fock space with nearly optimal cardinality.

Marcinkiewicz-Zygmund inequalities imply sampling
We first formulate a few properties of the distribution of Marcinkiewicz-Zygmund families. Consequently, every Λ n ∩B σn is a union of at most L separated sets with uniform separation δ > 0 independent of n.
For completeness we mention that the number of points in the transition region This can be shown as above by testing against the monomial z n . Before stating our main theorem, we recall that a sequence of sets Λ n ⊆ C converges weakly to Λ ⊆ C, if for all compact disks B ⊆ C where d(·, ·) denotes the Hausdorff distance between two compact sets in C. If every Λ n is the union of at most K uniformly separated sets with fixed separation δ, then with multiplicities µ(λ) ∈ {1, . . . , K}.
Theorem 5.2. Assume that (Λ n ) is a Marcinkiewicz-Zygmund family for the polynomials P n in F 2 . Let Λ be a weak limit of (Λ n ) or of some subsequence (Λ n k ). Then Λ is a sampling set for F 2 . Proof.
The assumption that Λ n is a Marcinkiewicz Zygmund family for P n in F 2 means that there exist A, B > 0 such that A p F 2 ≤ λ∈Λn |p(λ)| 2 kn(λ,λ) ≤ B p 2 F 2 for all polynomials p ∈ P n .
(i) Let B =B(w, ρ) be a closed disc. By Lemma 5.1(ii) #(Λ n ∩ B(w, ρ)) ≤ C for some constant C independent of n and B, provided that n is big enough. Since Λ is a weak limit of Λ n , we know that #(Λ ∩B(w, ρ)) ≤ C. This means that Λ is a union of K uniformly separated sets with separation δ > 0.
(ii) It follows immediately from (13) that Λ satisfies the upper bound in the sampling inequality for F 2 .
(iii) Lower bound. Fix a polynomial p ∈ P N (of degree N) and choose r > 0 such that where c δ is the constant in (12) for separation δ. To avoid the ugly notation in subscript, we write ν = r/π, ρ n = n/π, and σ n = n(1 − ε)/π. For p ∈ P N the Marcinkiewicz-Zygmund inequalities are satisfied for every n ≥ N, therefore If |λ| ≤ σ n , then k n (λ, λ) ≥ 1 2 k(λ, λ) = 1 2 e π|λ| 2 as a consequence of Lemma 2.1 and Proposition 3.1(v) Thus in the expressions for A n and B n we may replace the kernel k n for polynomials by the kernel k(z, z) = e π|z| 2 for Fock space. Consequently Since in this sum all points λ lie in the compact setB(0, ν+δ), the weak convergence (including multiplicities m(λ) ∈ {1, . . . , K}) implies the convergence to Λ and lim n→∞ A n ≤ 2 lim n→∞ λ∈Λn,|λ|≤ν+δ For the term B n , we recall that every Λ n ∩ B σn is a finite union of at most K uniformly separated sequences with separation δ and apply the tail estimate (13). Our choice of r and ν = r/π yields To treat C n , recall that p has degree N < n. We use the trivial estimate and substitute into C n to obtain By Lemma 5.1(ii) #Λ n ∩ B c σn ≤ B(n+1) (1−ε) n , whereas the ratio of the different reproducing kernels is k N (z, z) k n (z, z) = e π|z| 2 Γ(N + 1, π|z| 2 )n! e π|z| 2 Γ(n + 1, π|z| 2 )N! .
Combining the estimates for A n , B n , and C n and letting n go to ∞, we obtain the lower sampling inequality As the multiplicities satisfy 1 ≤ m(λ) ≤ K for λ ∈ Λ, we may omit them by changing the lower sampling constant to A/(2K).
Since polynomials are dense in F 2 , this estimate extends to all of F 2 .

Uniform Interpolation
In a sense the dual problem to sampling is the interpolation of function values. A set Λ ⊆ C is interpolating for F 2 , if for every a = (a λ ) λ∈Λ ∈ ℓ 2 (Λ) there exists f ∈ F 2 , such that f (λ)e −π|λ| 2 /2 = a λ . Equivalently, the set of normalized reproducing kernels κ λ = k λ / k λ F 2 = k λ /k(λ, λ) 1/2 is a Riesz sequence, i.e., there exists A, B > 0, such that for all a ∈ ℓ 2 (Λ). It suffices to require (17) only for all a with finite support. In analogy to Marcinkiewicz-Zygmund families for sampling, we define uniform families for interpolation as follows. We denote the normalized reproducing kernels in P n by κ n,λ = k n,λ / k n,λ F 2 . Definition 6.1. A sequence of finite sets Λ n ⊆ C is a uniform interpolating family for P n in F 2 , if there exist constants A, B > 0 independent of n, such that for n large enough, n ≥ n 0 , for all a ∈ ℓ 2 (Λ n ) .
Equivalently, for every a ∈ ℓ 2 (Λ n ) there exists a polynomial p ∈ P n , such that A further equivalent condition is that the associated Gram matrix with entries G µ,λ = κ n,λ , κ n,µ has the smallest eigenvalues λ min ≥ A [18, §2.3 Lem. 2]. The relation between sets of interpolation for F 2 and uniform interpolating families is similar to the case of sampling. Theorem 6.1. Assume that Λ ⊆ C is a set of interpolation for F 2 . For τ > 0 and ǫ > 0 set ρ n , such that πρ 2 n = n − √ n( √ 2 log n + τ ). Then for every τ > 0 large enough, the sets Λ n = Λ ∩ B ρn form a uniform interpolating family for P n in F 2 .
Combining these observations, we arrive at By choosing τ n = √ 2 log n + τ , with τ > 0 large enough, we achieve E op ≤ A/4 for n ≥ n 0 . As we have seen, this suffices to conclude that κ n,λ is a Riesz sequence in P n with lower constant independent of the degree n.
Similar to the case of Marcinkiewicz-Zygmund families for sampling, we obtain uniform families for interpolation with the correct cardinality.
Proof. The proof is similar to the one of Corollary 4.2. Theorem 6.3. Assume that (Λ n ) is a uniform interpolating family for the polynomials P n in F 2 . Let Λ be a weak limit of (Λ n ) or of some subsequence (Λ n k ). Then Λ is a set of interpolation for F 2 .
Proposition 6.4. There is no Marcinkiewicz-Zygmund family (Λ n ) for P n in F 2 with #Λ n = n + 1.

Proof.
A set Λ n with n+1 points is both sampling and interpolating for P n with the same constants for interpolation as for sampling. By Theorem 5.2 any weak limit Λ of a Marcinkiewicz-Zygmund family is a sampling set for F 2 , and by Theorem 6.3 Λ is a set of interpolation for F 2 . This is a contradiction, since F 2 does not admit any sets that are simultaneously sampling and interpolating. See, e.g., [23, Lemma 6.2].

Gabor frames for subspaces spanned by Hermite functions
By using the well-known connection between sampling in Fock space and the theory of Gaussian Gabor frames we may rephrase the main results in the language of Gabor frames for subspaces.
Recall that the Bargman transform is defined to be for z ∈ C. It maps functions and distributions on R to entire functions. We use the following properties of the Bargman transform. See e.g., [8].
(i) The Bargman transform is unitary from L 2 (R) onto Fock space F 2 .
(ii) Let φ z (t) = e −2πiyt e −π(t−x) 2 denote the time-frequency shift of the Gaussian by z = x + iy. Then Bφ z (w) = k z (w) = e πzw is the reproducing kernel of F 2 .
(iii) B maps the normalized Hermite functions h k , to the monomials e k (z) = π k k! 1/2 z k . With the Bargman transform all questions about the spanning properties of time-frequency shifts φ z of the Gaussian can be translated into questions about the reproducing kernels k z in Fock space. For instance, {φ λ : λ ∈ Λ} is a frame for L 2 (R), if and only if Λ is a sampling set for F 2 . Almost all statements about Gaussian Gabor frames have been obtained via complex analysis methods, notably the complete characterization of Gaussian Gabor frames by Lyubarski [17] and Seip [23] and many subsequent detailed investigations [6,7]. To this line of thought we add a statement about Gabor frames for distinguished subspaces spanned by Hermite polynomials. Constructions of this type have been used in signal processing [14].