Sharp results on sampling with derivatives in shift-invariant spaces and multi-window Gabor frames

We study the problem of sampling with derivatives in shift-invariant spaces generated by totally-positive functions of Gaussian type or by the hyperbolic secant. We provide sharp conditions in terms of weighted Beurling densities. As a by-product we derive new results about multi-window Gabor frames with respect to vectors of Hermite functions or totally positive functions.


Introduction and results
In the problem of sampling with derivatives one tries to recover or approximate a function by sampling a number of its derivatives. In analogy to Hermite interpolation this procedure is sometimes called Hermite sampling. For a well-defined problem one must fix a suitable signal model, which in engineering is usually a space of bandlimited functions (the Paley-Wiener space in mathematical terminology). In recent years the more general model of shift-invariant spaces has received considerable attention as a viable substitute for bandlimited functions. See [4] for an early survey.
Hermite sampling can be seen as a purely mathematical problem in approximation theory, but it is also informed by practical considerations. Whereas a sample f (λ) at a sampling point λ gives its pointwise value, the derivative f ′ (λ) measures the trend of f at λ, and higher derivatives yield information about the local approximation by Taylor polynomials. In addition, by taking several measurements at each point, one may hope to use fewer sampling points.
We study the problem of sampling with multiplicities in the shift-invariant space generated by a totally-positive function of Gaussian-type, where 1 ≤ p ≤ ∞. To describe the sampling process, we fix a sampling set Λ ⊆ R and a multiplicity function m Λ : Λ → N, and call (Λ, m Λ ) a set with multiplicity. The number m Λ (λ) indicates how many derivatives are sampled at λ ∈ Λ. We then say that (Λ, m Λ ) is a sampling set for V p (g) with 1 ≤ p < ∞, if there exist constants A, B > 0 such that If p = ∞, a sampling set is defined by the inequalities From a theoretical point of view the sampling inequality (1.2) completely solves the (Hermite) sampling problem. We note that a sampling inequality always leads to a general reconstruction algorithm based on frame theory [11]. In addition, for localized generators the frame algorithm converges even in the correct L p -norm [10]. Thus (1.2) is also a first step towards the numerical treatment of the sampling problem.
Our objective is the characterization of sampling sets satisfying the sampling inequality (1.2) and to obtain sharp conditions on the sampling set. In Beurling's tradition of complex analysis we will characterize sampling sets in terms of a weighted version of Beurling's lower density Within this setting we can already formulate our main result. Theorem 1.1. Let g be a totally positive function of Gaussian type. Let Λ ⊆ R be a separated set and let m Λ : Λ → N be a multiplicity function such that sup λ∈Λ m Λ (λ) < ∞.
Theorem 1.1 extends one of the results in [15] to sampling with multiplicities. We also have an analogous density result for the shift-invariant space generated by the hyperbolic secant. Theorem 1.2. Let ψ(x) = sech(ax) = 2 e ax +e −ax be the hyperbolic secant. Let Λ ⊆ R be a separated set and m Λ be a multiplicity function such that sup λ∈Λ m Λ (λ) < ∞.
For comparison, we state the corresponding sampling result for the Paley-Wiener space The statement is analogous to Theorems 1.1 and 1.2 and is considered folklore among complex analysts (we tested it!). Theorem 1.3. Let Λ ⊆ R be a separated set and let m Λ be a multiplicity function such that sup λ∈Λ m Λ (λ) < ∞.
Although folklore, Theorem 1.3 does not seem to have been formulated explicitly in the literature. A very interesting result involving divided differences of samples was proved for the Bernstein space PW ∞ by Lyubarski and Ortega-Cerda [18]. For the Fock space a result similar to Theorem 1.3 was derived early on by Brekke and Seip [9]. Theorems 1.1 and 1.2 have also several consequences for Gabor systems. Specifically, we characterize semi-regular sets Λ × βZ that generate a multiwindow Gabor frame with respect to the first n Hermite functions or with respect to a specific finite set of totally positive functions. See Section 6 for the precise formulations.
In the literature most sampling results for shift-invariant spaces work with the assumption that the sampling set Λ is "dense enough". However, when the sufficient density is made explicit, it is usually very far from the known necessary density, even in dimension 1. In fact, until [15] all authors use the covering density or maximum gap between samples, and the density then depends on some modulus of continuity of the generator. See [3] for one of the first nonuniform sampling theorems in shift-invariant spaces, [21] for nonuniform sampling with derivatives for bandlimited functions, and [2,5,23] for more recent examples of sufficient conditions for Hermite sampling in terms of the covering density.
In the light of [15] the sharp results for sampling with derivatives are perhaps not surprising, but they definitely go far beyond the current state-of-the-art. Our main point is to show the usefulness and power of the established methods, which consist of the combination of Beurling's techniques, spectral invariance, complex analysis, and the comparison of zero sets in different shift-invariant spaces. We believe that these methods carry a high potential in many other situations.
To arrive at sharp results, we combine several techniques. Roughly, we proceed in three steps: (i) We use Beurling's method of weak limits and show that the sampling inequality (1.3) for p = ∞ is equivalent to the fact that every weak limit of integer translates of Λ is a uniqueness set for V ∞ (g). In this way we obtain a general characterization of sampling sets without inequalities (Theorem 3.4).
(ii) To switch between sampling inequalities for p = ∞ and p < ∞, we use the theory of localized frames and Sjöstrand's beautiful version of Wiener's Lemma for convolution-dominated matrices [24]. These two steps are part of a general mathematical formalism that can be applied to many different situations. In particular, they work for shift-invariant spaces with almost arbitrary generators.
(iii) The concrete understanding then rests on the analysis of uniqueness sets for a particular shift-invariant space V p (g), or in other words, we need to analyze the zero sets of arbitrary functions in V p (g). For instance, for the classical Paley-Wiener space this is the relation between the density of the zero set of an entire function and its growth. This is precisely the aspect where we develop new arguments. Firstly, we observe that every function in V p (φ) for a Gaussian generator φ possesses an extension to an entire function, and secondly, we can relate the real zeros of some f ∈ V p (φ) to the complex zeros of its analytic extension. A similar, but technically more involved strategy works for the hyperbolic secant ψ(x) = (e ax − e −ax ) −1 . In a final step we relate the zero sets of functions in different shift-invariant spaces to each other. In this way we develop a direct line of arguments and avoid the detour in [15] via the characterization of Gaussian Gabor frames.
The paper is organized as follows: Section 2 introduces the necessary definitions for sampling in vector-valued shift-invariant spaces. These provide a convenient language to formulate the problem of sampling with derivatives. Section 3 then contains the main structural characterization of sampling with derivatives and the necessary density condition (Proposition 3.7). Section 4 is devoted to the investigation of the density of zero sets in shift-invariant spaces. This is the part that contains the main arguments and new proof ideas. The proofs of Theorems 1.1, 1.2, and 1.3 are then in Section 5. In Section 6 we draw some consequences of the sampling theorems with derivatives for multi-window Gabor frames. Finally Section 7 contains some of the postponed proofs of the structural results in Sections 2 and 3. As these are essentially known, we explain only the necessary modifications.
2. Vector-valued shift-invariant spaces and sampling 2.1. Vector-valued shift-invariant spaces. The treatment of sampling with derivatives requires us to formulate several standard concepts for vector-valued functions. In this section, we collect the precise definitions. For the proper formulation of sampling results we make use of the Wiener amalgam space W 0 = W 0 (R), which consists of continuous functions g such that Let G = (G 1 , . . . , G N ) ∈ (W 0 (R)) N . We consider the vector-valued shift-invariant space as a subspace of (L p (R)) N with norm and (F 1 , . . . , F N ) ∞ = max j=1,...,N F j ∞ . We always assume that G has stable integer shifts, i.e.
2.2. Sampling and weak limits. We consider tuples of sets Λ = (Λ 1 , . . . , Λ N ) with Λ j ⊆ R. We say that Λ is a sampling set for For p = ∞ the condition reads as F ∞ ≍ max j=1,...,N F j |Λ j ∞ . We say that Λ is a uniqueness set for V p (G) if whenever F ∈ V p (G) is such that F j ≡ 0 on Λ j , for all j = 1, . . . , N, then F ≡ 0. Clearly, sampling sets are also uniqueness sets.
We first recall Beurling's notion of a weak limit of a sequence of sets. A sequence {Λ n : n ≥ 1} of subsets of R is said to converge weakly to a set Λ ⊆ R, denoted Λ n w − → Λ, if for every open bounded interval (a, b) and every ε > 0, there exist n 0 ∈ N such that for all n ≥ n 0 Λ n ∩ (a, b) ⊆ Λ + (−ε, ε) and Λ ∩ (a, b) ⊆ Λ n + (−ε, ε).
We let W Z (Λ) denote the class of all sets Γ that can be obtained as weak limits of integer translates of Λ, i.e., Γ ∈ W Z (Λ) if there exists a sequence {k n : n ≥ 1} ⊆ Z such that Λ+k n w − → Γ. We extend this notion to tuples of sets as follows. Given two N-tuples of sets Λ = (Λ 1 , . . . , Λ N ) and Γ = (Γ 1 , . . . , Γ N ), we say that Γ ∈ W Z ( Λ) if there exists a sequence {k n : n ≥ 1} ⊆ Z such that Λ j + k n w − → Γ j for all 1 ≤ j ≤ N. (Note that the limits involve the same sequence {k n : n ≥ 1} for all j.) The following is a vector-valued extension of [15, Theorem 3.1].
) N have stable integer shifts and let Λ = (Λ 1 , . . . , Λ N ) be a tuple of separated sets. Then the following are equivalent.
The proof is similar to the scalar-valued version; a sketch of the proof is given in Section 7.

Sampling with multiplicities
3.1. Sets with multiplicities and derivatives. For N ∈ N we let W N 0 = W N 0 (R) be the class of functions g having derivatives up to order N − 1 in W 0 (R). For a set with multiplicity (Λ, m Λ ), we define its height as sup λ m Λ (λ). When sampling in shift-invariant spaces with generators on W N 0 (R) we assume that the sampling sets have height ≤ N. The lower density of (Λ, m Λ ) is defined by (1.4).

3.2.
Sampling with derivatives. We now describe how the problem of sampling with multiplicities can be reformulated in terms of sampling of vector-valued functions.
Let a generator g ∈ W N 0 (R) with stable integer shifts be given. We define G ∈ (W 0 (R)) N by choosing as components the derivatives of g, so There is an obvious one-to-one correspondence between In addition, since g has stable integer shifts, we have the norm equivalence This shows that G has stable integer shifts in the sense of (2.2).
Second, given a set with multiplicity (Λ, m Λ ) and height at most N < ∞, we consider the tuple sets Λ = (Λ 1 , . . . , Λ N ) given by Note that Λ 1 = Λ. The connection between vector-valued sampling and sampling with derivatives is stated in the following lemma, which is a direct consequence of our notation.
Finally, we interpret a weak limit Γ ∈ W Z ( Λ) as a set with multiplicity by setting Γ := Γ 1 and In order to keep our notations consistent, we also write (Γ, m Γ ) ∈ W Z (Λ, m Λ ) for the current situation.
A proof of Proposition 3.2 is given in Section 7. As a consequence, we obtain the following lemma; see, e.g. [15,Lemma 7.1] for a proof without multiplicities.

3.3.
Characterization of sampling with derivatives. Theorem 2.1 can be recast in terms of sampling with derivatives.
Theorem 3.4. Let g ∈ W N 0 (R) have stable integer shifts and let (Λ, m Λ ) be a separated set with multiplicity and height at most N < ∞. Then the following are equivalent. Theorem 3.5. Let (Λ, m Λ ) be a separated set with multiplicity and finite height. Then the following are equivalent.
As a replacement for the L 2 part of Theorem 3.4, we have the following result. Proposition 3.6. Let (Λ, m Λ ) be a separated set with multiplicity and finite height, and assume that (Λ, m Λ ) is a sampling set for PW ∞ . Then, for every α ∈ (0, 1), (αΛ, m Λ ) is a sampling set for PW 2 .
Theorem 3.5 and Proposition 3.6 are due to Beurling [7,8] (without multiplicities) -see also [19,Theorem 2.1]. A slight modification of the arguments yields the case with multiplicities.
3.4. Necessary density conditions. Proposition 3.7. Let g ∈ W N 0 (R) have stable integer shifts and let (Λ, m Λ ) be a separated set with multiplicity and height at most A similar statement holds for the Paley-Wiener space PW 2 .
Proposition 3.7 follows from standard results on density of frames, see e.g. [6,12]. See Section 7 for a sketch of a proof.

Density of zero sets in shift-invariant spaces
We derive sharp upper bounds for the density of real zeros of functions in shiftinvariant spaces with special generators. First, we use methods of complex analysis when the generator is a Gaussian (Section 4.1) or a hyperbolic secant (Section 4.2). The results and arguments are similar for both cases, but the case of the hyperbolic secant requires considerably more work and the analysis of meromorphic functions. In Sections 4.3 and 4.4 we then analyze the zero sets in shift-invariant spaces generated by a totally positive function of Gaussian type by means of a comparison theorem.
possesses an extension to an entire function satisfying the growth estimate Proof. Using e −a(x+iy−k) 2 = e ay 2 e −2aixy e 2aiky e −a(x−k) 2 , we obtain that (4.2) f (x + iy) = e ay 2 e −2aixy k∈Z c k e 2aiky e −a(x−k) 2 . Consequently and we may take Our key observation relates the real zeros of f ∈ V ∞ (φ a ) to the zeros of its analytic extension.
and λ ∈ R be a zero of f with multiplicity m. Then for every l ∈ π a Z, λ + il is a zero of the analytic extension of f with the same multiplicity m. In particular, if f (j) (λ) = 0 for j = 0, . . . , m − 1, then f (j) (λ + il) = 0 for j = 0, . . . , m − 1 and all l ∈ π a Z.
Proof. By (4.2) we obtain that because e 2aikl = 1 for all l ∈ π a Z. For higher multiplicities we argue as follows. Note first that d j dx j (e −ax 2 ) = p j (x)e −ax 2 for a polynomial of degree j satisfying the recurrence relation p j+1 ( . It follows that the set {p j : j = 0, . . . , m − 1} is a basis for the polynomials of degree smaller than m. Now assume that f ∈ V ∞ (φ a ) and f (j) (λ) = 0 for j = 0, . . . , m − 1. Then This implies that for every polynomial q of degree < m We now proceed as in (4.2) and find that, for j = 0, . . . , m − 1, Note that e 2aikl = 1 for all l ∈ π a Z. We insert the Taylor expansion of p j at λ − k, i.e., and we obtain that Since each p (r) j is a polynomial of degree < m, (4.3) implies that f (j) (λ + il) = 0 for all l ∈ π a Z and j = 0, . . . , m − 1. This shows that the multiplicity of λ + il is at least that of λ. Reversing the roles of λ and λ + il we see that the multiplicities are actually equal.
We recall Jensen's formula, which relates the number of zeros n(r) in a disk B(0, r) to the growth of an entire function by the identity   Proof. Note that N f = {λ ∈ R : f (λ) = 0} is the set of real zeros of f . By Lemma 4.2, the set of complex zeros of (the analytic extension of) f contains the set N f + i π a Z ⊆ C, and, moreover, multiplicities are preserved. To prove the theorem, we argue indirectly and assume that D − (N f , m f ) > 1. Then there exists ν > 1 and R 0 , such that Let n(r) be the number of complex zeros of f inside the open disk B(0, r) ⊆ C counted with multiplicities. Let us assume for the moment that f (0) = 0.
The right-hand side of Jensen's formula (4.4) can be estimated, by means of the growth estimate (4.1), as To estimate the left-hand side of (4.4), we choose R ∈ N and R ≥ R 0 and partition [−R 2 , R 2 ) = R−1 k=−R [kR, (k + 1)R). On each interval there are at least νR real zeros of f counted with multiplicity. By symmetry it is enough to consider intervals [kR, (k + 1)R) with 0 ≤ k ≤ R − 1. By Lemma 4.2, for each real zero λ ∈ [kR, (k + 1)R), with a certain multiplicity m, there are 2⌊ a π (R 2 ) 2 − λ 2 ⌋ + 1 ≥ 2a π R 4 − (k + 1) 2 R 2 − 1 complex zeros λ + il, l ∈ π a Z in the disk B(0, R 2 ), each with multiplicity m. By counting with multiplicities, there are at least By summing over (positive and negative) k, we obtain the following lower bound for the number of complex zeros of f in B(0, R 2 ): The last sum is a Riemann sum of the integral ) > 1. Then, for some R 2 ≥ R 1 and all R ≥ R 2 , we conclude that or, equivalently, n(r) ≥ aβr 2 for r ≥ R 2 2 .
Therefore, the left-hand side of (4.4) can be estimated as Since β > 1, this estimate is incompatible with the growth of f as encoded in (4.5). Therefore D − (N f , m f ) > 1 is impossible. This concludes the proof for f such that f (0) = 0. If f (0) = 0, we let n ≥ 0 be the vanishing order of f at 0 and apply the previous argument tof (z) := z −n f (z). Alternatively, one can verify directly that if f ≡ 0, then there exists k ∈ Z such that f (k) = 0, and considerf (x) = f (x + k) with f ∈ V ∞ (φ).

4.2.
The hyperbolic secant. Let ψ a (x) = sech(ax) = 2 e ax +e −ax . Our goal is to study the shift-invariant space generated by ψ a . While in [15] we studied V 2 (ψ a ) by exploiting a connection to Gabor analysis, and a certain representation of the Zak transform of ψ a due to Janssen and Strohmer [17], here we consider meromorphic extensions of the functions in V ∞ (ψ a ). We introduce the following notation. For real x we denote the roundoff error to the nearest integer as {x} := x − l, where l ∈ Z and {x} ∈ [−1/2, 1/2).  shows that |ψ a (x − k + iy)| e −a|x−k| , if |x − k| ≥ 1 and y is arbitrary. We consider the covering of C given by U s,t := x + iy ∈ C : |x − s| < 3/4, |y − π a (t + 1/2)| < 3π 4a , s, t ∈ Z.
On U s,t , the partial sums have at most a simple pole at s + iπ a ( 1 2 + t) and are otherwise analytic. Since, for x + iy ∈ U s,t , the partial sums f N converge uniformly on U s,t \ s + iπ a ( 1 2 + t) to an analytic extension of f . More precisely, (Note that this is stronger than the usual uniform convergence on compact sets.) This fact implies that f has at most a simple pole at z = s + iπ a (1/2 + t). Hence, f is meromorphic on C with at most simple poles in Z + iπ a 1 2 + Z . For the growth estimate (4.6) we let x + iy ∈ C \ P f and write x = l + {x} with l ∈ Z and {x} ∈ [−1/2, 1/2). Then we have For all k = l we observe that |x − k| ≥ |l − k| − |{x}| ≥ 1 2 ≥ |{x}|. Therefore, we have sinh 2 a(x − k) ≥ sinh 2 a {x}. Since the rational function r(y) = c+y d+y with d ≥ c ≥ 0, d > 0 is increasing for y > 0, we obtain that, for all k = l, Therefore, we have This proves the first inequality in (4.6). For the second inequality note that |sinh ax| ≥ |ax| for all x ∈ R and, by periodicity and elementary trigonometric identities, |cos ay| = |sin π ay π − 1 2 | ≥ 2| ay π − 1 2 | for all y ∈ R. Hence, we obtain |ψ a ({x} + iy)| = sinh 2 a {x} + cos 2 ay −1/2 ≤ min{|a {x}| −1 , |2 ay π − 1 2 | −1 }, which gives the second inequality in (4.6).
The following result is an analogue of Lemma 4.2 for V ∞ (ψ a ). Lemma 4.5. Let f ∈ V ∞ (ψ a ) and λ ∈ R be a zero of f with multiplicity m. Then for every l ∈ π a Z, λ + il is a zero of the meromorphic extension of f with the same multiplicity m.
Proof. For every x ∈ R and l = πt a ∈ π a Z we have cosh a(x + il) = cosh ax cos al + i sinh ax sin al = (−1) t cosh ax.
This implies that the Taylor expansions of f around z 0 = x ∈ R and around z l = x + il have exactly the same coefficients, up to a factor (−1) t . In particular, f (j) (λ) = 0 holds for some λ ∈ R and j ≥ 0 if and only if f (j) (λ + il) = 0 for all l ∈ π a Z.     Proof. We divide the integral into four pieces corresponding to For θ ∈ I 0 ∪ I 2 , we let By (4.6), we have log|f (re iθ )| ≤ log(C c ∞ ) − log|2 ay π − 1 2 | and (using dθ = ±dy/ r 2 − y 2 ) Note that log|2 ay π − 1 2 | ≤ 0 and r 2 − y 2 ≥ r/ √ 2 for all y ∈ − r √ 2 , r √ 2 . Therefore, For the last integral, we use the substitution u = ay π and observe that the resulting integrand is even and periodic with period 1. This gives for all c < d and finally In the same way, for θ ∈ I 1 ∪ I 3 we let and obtain from (4.6) The same techniques as before give Hence, we obtain Combining both integrals, we get for r ≥ 1 1 2π which is bounded for r ≥ 1. This completes the proof.

4.3.
Transference of zero sets. The following lemma modifies [15, Lemma 5.1] to include multiplicities and allows us to compare the density of zero sets in different shift-invariant spaces.
Lemma 4.8. Let f ∈ C ∞ (R) be real-valued and m f : N f → N be the multiplicity function of its zeros. For a ∈ R let g = aI + d dx f . Then For f ∈ C N −1 (R) the same statement holds, replacing m f and m g by the multiplicity functions of the zeros of height at most N and N − 1, respectively.
Proof. Let f ∈ C ∞ (R). Note that aI + d dx = e −ax d dx e ax . We define h ∈ C ∞ (R) by h(x) = e ax f (x) and note that N h = N f with equal multiplicities m h = m f . Furthermore, since Since this holds for every finite subset as claimed. Finally, for f ∈ C N −1 (R), g ∈ C N −2 (R), and the same argument applies to the multiplicity functions of zeros of height N and N − 1.
Although generically one would expect equality in (4.10), the density of the zero set may actually jump. Let  In particular, if D − (Λ) > 1, then Λ is a uniqueness set for V ∞ (g).
Proof. The proof is an adaption of the argument in [15] using multiplicities. Recall that g is real-valued and has stable integer shifts. Let c ∈ ℓ ∞ (Z) and assume that Since g is real-valued, we may assume without loss of generality that f is also real-valued (by replacing c k by ℜ(c k ) or ℑ(c k ) if necessary). Using (1.1), write (1 + 2πiδ j ξ) −1φ (ξ), δ 1 , . . . , δ n ∈ R \ {0}, c > 0, whereφ(ξ) = e −cξ 2 . In other words, φ = n j=1 I + δ j d dx g is a Gaussian. Since φ, g, and their derivatives decay exponentially, we may interchange summation and differentiation in f , and obtain that The repeated use of Lemma 4.8 implies that Hence, by Theorem 4.3, h = k c k φ(· − k) ≡ 0. Hence c k ≡ 0 and f ≡ 0, as claimed.
4.5. Bandlimited functions. For a simple comparison of the results in Theorems 4.3, 4.6, and 4.9, we mention the following result for bandlimited functions.
Proof. The result follows from the Paley-Wiener characterization of bandlimited functions as restrictions of entire functions of exponential type, and Jensen's formula. Beurling's proof [7,8] applies almost verbatim.

Proof of the sampling theorems
The proofs of our main theorems are now short and follow from the combination of the characterization of sampling sets without inequalities (Theorem 3.4) and the new insights about the density of zero sets in shift-invariant spaces (Section 4).
Proof of Theorem 1.2. The proof is the same as for Theorem 1.1; this time we resort to Theorem 4.6 (instead of Theorem 4.9).

Consequences for Gabor frames
The Hermite-sampling results of Theorems 1.1 and 1.2 can be applied in order to obtain sharp density results for multi-window Gabor frames. This extends our previous work in [15] and was, in fact, one of our original motivations for the present work. We obtain new families of multi-window Gabor frames with optimal conditions for semi-regular sets of time-frequency shifts.
6.2. Connection between sampling and Gabor frames. For semi-regular sets ∆, the Gabor frame property can be related to a sampling problem as follows.
Let Γ ∈ W Z ( Λ + x). This set is necessarily of the form Γ = (Γ, . . . , Γ), for some Γ ∈ W Z (Λ + x), and, by Lemma 3.3, D − (Γ) > 1/N. Assume that F ∈ V ∞ (G) vanishes on Γ. We need to show that F ≡ 0. Explicitly F is given by an expansion We now relate the sampling problem for vector-valued functions to a sampling problem with derivatives. To do this, we set P = (p 1 , . . . , p N ) and Q = (1, x, . . . , x N −1 ). By assumption on P , there is an invertible N × N-matrix B, such that BP = Q, i.e., x j−1 = N k=1 b jk p k (x) for j = 1, . . . , N and thus

Consequently, after taking linear combinations of translates we obtain
where f = l c l g(· − l) ∈ V ∞ (g) is the first component of BF . If F vanishes on Γ, then also f (j−1) vanishes on Γ for j = 1, . . . , N. Hence, f vanishes on Γ with multiplicity N and D − (N f , m f ) ≥ ND − (Γ) > 1. By Theorem 4.9 or 4.6, this implies that f ≡ 0. Hence, c k ≡ 0 and F ≡ 0, as desired.
We single out two special cases of Theorem 6.2. For the second corollary we use the basis of Hermite functions {h k : k ≥ 0} which is defined by , with the Hermite polynomials H k of degree k and some normalizing constant γ k > 0.
is. Because the Hermite polynomials H k , k = 0, . . . , N − 1, form a basis for the polynomials of degree < N, the span of h k , 0 ≤ k ≤ N − 1, is the same as the span of all derivatives d j dx j e −πx 2 , 0 ≤ j ≤ N − 1. The result is a consequence of Theorem 6.2. Corollary 6.4 actually follows from a sampling result of Brekke and Seip in Fock space [9]. It can also be reformulated for spaces of polyanalytic functions. For this connection see [1]. 7. Postponed proofs 7.1. Proof of Proposition 3.2. For sets without multiplicities, i.e., m Λ ≡ 1, the proposition is classical.
Then Λ is a sampling set for V (G) if and only if A : ℓ p (Z) → ℓ p (I) is bounded below. The independence of p of this property for the range p ∈ [1, +∞] follows from (a slight extension of) Sjöstrand's Wiener-type lemma [24]. The formulation in [15, Proposition A.1] is applicable directly. Specifically, [15, Proposition A.1] concerns a matrix indexed by two relatively separated subsets of the Euclidean space (where a relatively separated set is just a finite union of separated sets). In our case, I is a relatively separated subset of R 2 , while Z can be embedded into R 2 as Z × {0}. This accounts for the equivalences (a) ⇔ (b). The other implications follow, with very minor modifications, as in the proof of [15,Theorem 3.1]. See also [13,Section 4] for some relevant technical tools. .

7.3.
Sketch of a proof of Proposition 3.7. The proposition follows from the theory of density of frames. The Paley-Wiener case is explicitly treated in [14] following the technique of Ramanathan and Steger [20]. For shift-invariant spaces with generators in g ∈ W N 0 (R), we can use the abstract density results for frames from [6] as follows.