Gaussian Gabor frames, Seshadri constants and generalized Buser--Sarnak invariants

We investigate the frame set of regular multivariate Gaussian Gabor frames using methods from K\"ahler geometry such as H\"ormander's $\dbar$-$L^2$ estimate with singular weight, Demailly's Calabi--Yau method for K\"ahler currents and a K\"ahler-variant generalization of the symplectic embedding theorem of McDuff--Polterovich for ellipsoids. Our approach is based on the well-known link between sets of interpolation for the Bargmann-Fock space and the frame set of multivariate Gaussian Gabor frames. We state sufficient conditions in terms of a certain extremal type Seshadri constant of the complex torus associated to a lattice to be a set of interpolation for the Bargmann-Fock space, and give also a condition in terms of the generalized Buser-Sarnak invariant of the lattice. In particular, we obtain an effective Gaussian Gabor frame criterion in terms of the covolume for almost all lattices, which is the first general covolume criterion in multivariate Gaussian Gabor frame theory. The recent Berndtsson--Lempert method and the Ohsawa--Takegoshi extension theorem also allow us to give explicit estimates for the frame bounds in terms of certain Robin constant. In the one-dimensional case we obtain a sharp estimate of the Robin constant using Faltings' theta metric formula for the Arakelov Green functions.


INTRODUCTION
Let Λ be a lattice in R 2n and (1.1) H := {Ω ∈ gl(n, C) : Ω = Ω T , Im Ω is positive definite} be the Siegel upper half-space. Fix Ω ∈ H, a Gaussian Gabor system, denoted by (g Ω , Λ), is the family of functions {π λ g Ω } λ∈Λ , where Λ is a lattice in R 2n and (π λ g Ω )(t) := e 2πiξ T t g Ω (t − x), λ := (ξ, x) ∈ Λ, ξ T t := n j=1 ξ j t j , denotes a time-frequency shift of the Gaussian g Ω (t) := e πit T Ωt . Letting I denote the identity matrix, we note that g iI (t) is the standard Gaussian e −π|t| 2 . The frame set of g Ω is the set of all lattices Λ in R 2n such that {π λ g Ω } λ∈Λ is a frame for L 2 (R n ). Here, by a frame, we mean there exist positive constants A, B (called frame bounds) such that Date: June 2, 2023. 1 where (·, ·) denotes the L 2 -inner product. Using the Bargmann transform, one may reformulate the frame property for Gaussian Gabor frames in terms of sampling property of the Bargmann-Fock space in complex analysis. In the one-dimensional case there is a density criterion for the interpolation problem in the Bargmann-Fock space due to Lyubarskii and Seip-Wallstén, which implies a seminal result in the theory of Gaussian Gabor frames. Namely, that {π λ g iI } λ∈Λ is a Gabor frame if and only if the covolume |Λ| < 1, [44,56,57]. Recent progress on the description of the frame set of a Gabor atom has been made for totally positive functions [29,28] and for rational functions [3]. Note that all aforementioned results on frame sets for Gabor systems are for uniformly discrete point sets in the plane. The generalization to the higherdimensional case has been one of the most intriguing problems in the study of Gaussian Gabor frames since the methods in [44,56,57] do not have natural counterparts in the theory of several complex variables. The reason being that the theory of sampling and interpolation in several complex variables [45,42,24,27,43] is far more intricate than in the one-dimensional case (see [53, section 3]) and despite considerable effort not well understood. In particular, the following central problem in multivariate Gaussian Gabor frame theory is still open.
Problem A: Is there an equivalent Gabor frame criterion for (g iI , Λ) only in terms of the covolume |Λ| for almost all lattices Λ in R 2n (n > 1)?
We obtain the following partial result, which is a direct consequence of Proposition 1.4 and our Hörmander criterion in section 1.2.
Our second main result is a multivariate Gaussian Gabor frame criterion for general lattices (see the remark after Theorem A in section 1.2 for the proof and Corollary 1.8 for applications).
Remark. In case (Ω, Λ) is transcendental the above theorem is equivalent to our first main theorem. In order to prove this equivalence, we generalize (see Theorem A in section 1.2) McDuff-Polterovich's result [46] (see Theorem 3.4) to all Kähler ellipsoid embeddings (see section 3.1).
In the one-dimensional case, we also obtain the following frame bound estimates (see Theorem B in section 1.3 for more results), which can be seen as an effective version of [10, Theorem 1.1]. Theorem 1.3 (Third main theorem). Let Λ be a lattice in R × R. Suppose that the lattice where η(τ ) := e πiτ /12 Π ∞ n=1 (1 − e 2πinτ ) is the Dedekind eta function. The above three main theorems are special cases of the Hörmander criterion (Theorem 1.7), Theorem A and B in section 1.2. The whole paper is organized as follows. Our first main theorem is an extension of the approach by Berndtsson-Ortega Cerdà [8] for one-dimensional Gaussian Gabor frames to the multivariate case by utilizing the theory of Hörmander's L 2 -estimates for ∂ in the higher-dimensional case, which has been developed during the past two decades and has received quite some attention [49,50,18,6,4,5,9,30,7]. The new idea is to apply Demailly's mass concentration technique [17] (see [61] for a nice survey).
A crucial notion in the proof of our second main theorem is a generalized extremal type Seshadri constant (see Definition 1.5 and Theorem 3.3 for the extremal property), which replaces the covolume of a lattice in the one-dimensional case. Note also that the Seshadri constant has a quite different flavor than the (Beurling) densities used in the discussion of Gabor frames, since it actually takes into account the volumina of all subvarieties of the complex torus (see Theorem 3.9) and not just of the whole complex torus.
Our third main theorem is based on a recent result of Berndtsson-Lempert [7], which also yields explicit estimates for the multivariate Gaussian Gabor frame bounds (see Theorem 5.1 and Theorem 5.2).
A short account of our other related results is given in section 1.
where O(C n ) denotes the space of holomorphic functions on C n and we omit the Lebesgue measure in the integral. Let Γ be a lattice in C n .
Denote by |Γ| (the covolume of Γ) the volume of the torus X := C n /Γ with respect to the Lebesgue measure. In the one-dimensional case, Lyubarskii, Seip and Wallstén [44,56,57] (see [52] for the most general one-dimensional generalization) proved that Theorem 1.5. A lattice Γ in C is a set of interpolation for F 2 if and only if |Γ| > 1.
The above result suggests the following definition.
Definition 1.4. A lattice Γ in C n is said to be transcendental if the only positive dimensional analytic subvariety of X := C n /Γ is X itself.
By Proposition 1.4 and Proposition 1.6, we know that the following criterion implies our first main result -Theorem 1.1.
Theorem 1.7 (The Hörmader criterion). Let Γ be a lattice in C n . If ι Γ > 1 then Γ is a set of interpolation for F 2 . In particular, if Γ is transcendental and |Γ| > n n n! , then Γ is a set of interpolation for F 2 .
The β-Seshadri constant of (X, ω) at x ∈ X is defined by where "ω-psh" means that ψ is upper semi continuous on X with in the sense of currents on X.
Remark. The above theorem implies Theorem 1.2. In fact, by the above theorem and (1.7), the assumption in Theorem 1.2 implies that ι Γ > 1. Hence our second main theorem follows from the Hörmander criterion, Theorem 1.7 and Proposition 1.4.
In case X = C n /Γ and ω = dd c (π|z| 2 ) is the Euclidean Kähler form, we know that B β r is included in X if and only if (γ + B β r ) ∩ B β r = ∅, ∀ 0 = γ ∈ Γ, which is equivalent to that Hence Theorem A gives Apply the Hörmander criterion above we get: Corollary 1.8. Let Γ be a lattice in C n . If sup β∈B inf 0 =z∈Γ n j=1 β j |z j | 2 > 4 π , then the Hörmander constant ι Γ > 1 (see Definition 1.3) and Γ is a set of interpolation for F 2 .
Remark. In case all β j are equal to 1/n, is known as the Buser-Sarnak invariant m(Γ) (see [12,40], [ 1.3. Part II: Gaussian Gabor frames. In this section we shall show how to apply the preceding results on sets of interpolation in F 2 in Gabor analysis. We use the same symbol Γ Ω,Λ • in (1.2) to denote the underlying lattice By a direct computation, we know that the symplectic dual of Γ Ω,Λ • is equal to Γ Ω,Λ := {(Im Ω) −1/2 (ξ + Re Ωx, Im Ωx) ∈ R n × R n : (ξ, x) ∈ Λ}.
Hence Proposition 1.4 gives the following: is a linear mapping preserving the standard symplectic form ω := dξ T ∧ dx on R n × R n . In dimension one, we know that (g Ω , Λ) defines a frame in L 2 (R) if and only if (g iI , Λ) defines a frame in L 2 (R) by the Theorem of Lyubarskii-Seip-Wallstén. However, the following result implies that this is not the case for multivariate Gabor systems.
Theorem 1.10. The Gabor system (g iI , (Z ⊕ 1 2 Z) 2 ) does not give a frame in L 2 (R 2 ). Note that we have (1) There exists an R-linear isomorphism f of R 4 preserving the standard symplectic form dξ T ∧ dx on R 2 × R 2 such that (g iI , f (Z ⊕ 1 2 Z) 2 ) does give a frame in L 2 (R 2 ); (2) There exists Ω ∈ H such that (g Ω , (Z ⊕ 1 2 Z) 2 ) does give a frame in L 2 (R 2 ). Moreover, the set {Ω ∈ H : (g Ω , (Z ⊕ 1 2 Z) 2 ) is not a frame} is included in a closed analytic subset of the Siegel upper half-space H.
In the one-dimensional case, we obtain the following estimates thanks to Faltings' Green function formula [21] (see also section 5.3).
Then |Λ| −1 = |Γ| ≥ C and we have the following estimates: (2) If 1 < C < 2 then for all f ∈ L 2 (R) with ||f || = 1, we have (3) The most general case. Suppose that Λ's symplectic dual lattice Γ in C is generated by {a, τ } with a > 0 and a Im τ > 1. Then for all f ∈ L 2 (R) with ||f || = 1, we have Remark. With the notation in (3), we have For Ω ∈ H (see (1.1)) we identify R n × R n with C n via (ξ, x) → z := ξ + Ωx. We call the short-time Fourier transform of f ∈ L 2 (R n ) with respect to g Ω (t) = e πit T Ωt the Gabor transform of f : The latter can be related to the Ω-Bargmann transform as follows: Since z = ξ + Ωx implies that we know that (2.1) gives Sometimes we shall omit the Lebesgue volume form i 2 ∂∂|z| 2 n in the above. Now we are ready to introduce the following definition.
We call the space of holomorphic functions F on C n with the Ω-Bargmann-Fock space and denote it by F 2 Ω . Remark. In case Ω = iI n , where I n denotes the identity matrix, F 2 Ω is precisely the following classical Bargmann-Fock space (see [47, page 7] for the related Von-Neumann-Stone theorem) In time-frequency analysis, the Bargmann-Fock space F 2 in (1.4) is more widely used. But these two spaces are naturally isomorphic to each other since Let us briefly recall the basics of Gaussian Gabor frames: We associate to the Gabor system (g Ω , Λ) the following operators: • An elementary computation shows that (C Λ g Ω ) * = D Λ g Ω and thus S Λ g Ω := (C Λ g Ω ) * • C Λ g Ω is a selfadjoint operator. The following result is well known, see [23].
Proof. We shall give a proof for readers' convenience. Using the Ω-Bargmann transform, it suffices to show that By the submean inequality, we know that the above inequality is true for There is a fundamental duality theory (see [15,38,55,13] and [36,Theorem 4.22]) that links the Gabor system (g Ω , Λ) with another Gabor system associated to the symplectic dual lattice/adjoint lattice defined in Definition 1.1.
There is an intricate link between Gabor analysis and Bargmann-Fock spaces: a Gaussian Gabor system (g Ω , Λ) is a frame if and only if Λ is a set of sampling for F 2 Ω , and (g Ω , Λ • ) is a Riesz basis for its closed linear span if and only if Λ • is a set of interpolation for F 2 Ω , see [27] for the standard case Ω = iI.
Motivated by this we introduce the following well known notions: (2) T is referred to as interpolating if T is surjective and there exist constants A, B > 0 such that where f c denotes the (unique) solution of T (·) = c with minimal norm.
The constants A, B above are called the sampling (interpolating) bounds.

Proposition 2.3. T is sampling with (2.4) if and only if T * is interpolating with (2.5).
Proof. Assume that T is sampling with (2.4). Then the eigenvalues of T * T lie in [A, B], thus T * T has an inverse, say S : which gives ||f || 2 ≤ B||T Sf || 2 . This establishes one direction and the other direction may be deduced in a similar manner.
The following theorem follows directly from Theorem 2.2 and Definition 2.2.
g Ω is sampling if and only (g Ω , Λ) defines a frame in L 2 (R n ). The density theorem for Gabor frames states that if C Λ g Ω is sampling, then |Λ| ≤ 1. Furthermore, a Balian-Low type theorem (see [1,Theorem 1.5] or [25] for related results associated to general Fock spaces) further gives: g Ω is sampling then |Λ| < 1. The above two theorems and Proposition 2.3 imply g Ω is sampling if and only if C Λ • g Ω is interpolation. Moreover, the interpolation bounds are a scalar multiple of the sampling bounds. In particular, C Λ g Ω can not be both sampling and interpolation. Proof. The first part follows directly from Theorem 2.4 and Proposition 2.3. For the second part, notice that if C Λ g Ω is both sampling and interpolation, we must have which is a contradiction since |Λ| · |Λ • | = 1.
Remark. In case n = 1 and for some complex number a with Im a > 0, we known that C Λ g is sampling if and only if |Λ| < 1 (see [44,56,57]). For general n, g Ω (t) := e πit T Ωt , Theorem 1.10 implies that there exists a lattice Λ in R n × R n such that C Λ g Ω is sampling (resp. interpolation) for some Ω ∈ H but not for all Ω ∈ H. On the other hand, if C Λ The duality principle Theorem 2.4 further implies: Theorem 2.7. With the notation in the above proof, the following statements are equivalent: (3) (Λ, g Ω ) defines a frame in L 2 (R n ) and for all f ∈ L 2 (R n ), ||f || = 1, By Proposition 2.3, the above inequality is equivalent to that Let φ be a plurisubharmonic function such that φ − δ|z| 2 is also plurisubhamonic on C n for some positive constant δ. Then there is a smooth function a on C n such that ∂u = v and where Proof of the Hörmander criterion (Theorem 1.7). Notice that the β-Seshadri constant does not depend on the choose of x ∈ C n /Γ. Thus, if the Hörmander constant is bigger than one then there exist γ > 1 and an ω euc -psh function ψ on C n /Γ such that ψ = γT β near 0 ∈ C n /Γ for some β ∈ B. Let p : C n → C n /Γ, be the natural quotient mapping. Fix c = {c λ } such that λ∈Γ |c λ | 2 e −π|λ| 2 = 1.
Let us apply Theorem 2.8 to where χ is a smooth function on R that is equal to 1 near the origin and equals to 0 outside a smooth ball of radius r. Let us take r such that Then we know that v is smooth, ∂v = 0 and C n |v| 2 e −φ < C, for some constant thay does not depend on the sequence c = {c λ }. Moreover, since ψ is ω eucpsh, we know that φ(z) − (1 − γ −1 )π|z| 2 is plurisubharmonic. Thus Theorem 2.8 implies that there exists a smooth function u such that ∂u = v and By a direct computation we know that e −T β is not integrable near 0 ∈ C n /Γ, hence e −φ is not integrable near Γ and (2.7) implies that u vanishes at Γ. Take for some constant C 1 does not depend on c (notice that ψ is bounded from above) and F (λ) = c λ for all λ ∈ Γ. Thus Γ is a set of interpolation. The final statement is a direct consequence pf Proposition 1.6, which will be proved in section 3.1.
In order to estimate the L 2 norm of the extension F in the above proof, we shall introduce the following Ohsawa-Takegoshi type theorem [49] proved by Berndtsson and Lempert (see [7,Theorem 3.8], the main theorem in [30] and [9] for related results). Theorem 2.9. Let Γ be a lattice in C n . Assume that there exists a non positive Γ invariant function ψ on C n such that ψ(z) + π|z| 2 is plurisubharmonic on C n , ψ is smooth outside Γ and ψ(z) − γ log |z| 2 is bounded near the origin for some constant γ > n. Then for every sequence of complex numbers {c λ } λ∈Γ with λ∈Γ |c λ | 2 e −π|λ| 2 = 1, there exists F ∈ F 2 such that F (λ) = c λ for all λ ∈ Γ and Proof. Since ψ is Γ invariant, we know that ψ has isolated order γ log poles at Γ, one may use Ohsawa-Takegoshi extension theorem to extend L 2 functions from Γ to C n . Denote by F the extension with minimal L 2 norm. By our assumption Hence [7,Theorem 3.8] or the main theorem in [30] implies thus our theorem follows.

Transcendental lattices and jet interpolations.
Let us first introduce the following definition for jet interpolations.
Definition 2.3. Let k ≥ 0 be an integer. Let Γ be a lattice in C n . Put We say that Γ is a set of k-jet interpolation for F 2 if there exists a constant C > 0 such that for every sequence of complex numbers {c λ,α } λ∈Γ,α∈N k with λ∈Γ,α∈N k |c λ,α | 2 e −π|λ| 2 = 1, there exists F ∈ F 2 with and ||F || 2 ≤ C.
The proof of the Hörmander criterion above also implies the following result.
Theorem 2.10. Let k ≥ 0 be an integer. Let Γ be a transcendental lattice in C n . Assume that then Γ is a set of k-jet interpolation for F 2 .
Proof. Since Γ is transcendental, by (3.2), we know that (2.8) implies that there exists a non positive Γ invariant function ψ on C n such that ψ(z) + π|z| 2 is plurisubharmonic on C n , ψ is smooth outside Γ and ψ(z) − γ log |z| 2 is bounded near the origin for some constant γ > n + k. Thus the Hörmander L 2 estimate with singular weight ψ (similar to the proof of the Hörmander criterion above) gives the above theorem.
Remark: If Γ is transcendental with |Γ| > n n n! then n + k n Γ > (n + k) n n! , thus we know that n+k n Γ is a set of k-jet interpolation for F 2 . In one dimensional case, we have the following theorem [26].
Theorem 2.11. Let Γ be a lattice in C. Then the followings are equivalent: (1) Γ is a set of interpolation for F 2 ; (2) √ k + 1 Γ is a set of k-jet interpolation for F 2 for some positive integer k; (3) |Γ| > 1.
For the higher-dimensional cases, we can prove the following result.
Theorem 2.12. Let Γ be a transcendental lattice in C n . If n+k n Γ is a set of k-jet interpolation for F 2 for some non-negative integer k then |Γ| ≥ (k+1) n (n+k) n n n n! .
Proof. Put ∇ α F := e π|z| 2 ∂ α (e −π|z| 2 F ) and Γ k := n+k n Γ. Assume that n+k n Γ is a set of k-jet interpolation for F 2 . Let us define Then by the Balian-Low type theorem, we know that G is not identically zero on C n . We claim that G is Γ k invariant. In fact, if we put Then Hence G is Γ k invariant. Put ψ = log G, we know that ψ is Γ invariant, ψ(z) + π|z| 2 is plurisubharmonic on C n and ψ(z) − (k + 1) log |z| 2 is bounded above near z = 0. Since Γ is transcendental, we know that Γ k is also transcendental, thus the Γ k invariant analytic set {ψ = −∞} is discrete. Hence (3.2) gives that (n!|Γ k |) 1/n ≥ k + 1, from which our theorem follows.

Remark:
The above theorem is our motivation for the conjecture A after Theorem 1.1, moreover, notice that lim k→∞ k+1 n+k = 1, the above theorem also suggests the following higher-dimensional analogue of Theorem 2.11.
Conjecture B: Let Γ be a transcendental lattice in C n . Then the followings are equivalent: (1) Γ is a set of interpolation for F 2 ; (2) n+k n Γ is a set of k-jet interpolation for F 2 for some positive integer k; (3) |Γ| > n n n! . Remark: From Proposition 1.4, we know that the above conjecture implies conjecture A.

HÖRMANDER CONSTANTS AND KÄHLER EMBEDDINGS
3.1. Hörmander constants and proof of Proposition 1.6. In Definition 1.3 and Definition 1.5 we have defined the Hörmander constants and the β-Seshadri constants for an n-dimensional compact Kähler manifold (X, ω). If all β j = 1/n and ω ∈ c 1 (L) for some ample line bundle L then we have where ǫ x (ω) denotes the Seshadri constant introduced by Demailly in [18] (in fact, from (6.2) in [18], we have nǫ x (ω; β) = γ(L, x), but Theorem 6.4 in [18] tells us that γ(L, x) is precisely the Seshadri constant used in algebraic geometry when L is ample). In general, the condition β j = 1 is used to make sure that sup{c ≥ 0 : e −cT β is integrable near z = 0} = 1.
Remark. For transcendental ω on a general compact Kähler manifold, we know that (see Theorem 3.2 below for the proof and generalizations) nǫ x (ω; β) is equal to the generalized Seshadri constant (also denoted by ǫ x (ω)) defined by Tosatti in [60, section 4.4]. In this general case, we shall prove the following result.
Proposition 3.1. Let (X, ω) be an n-dimensional compact Kähler manifold. Assume that X has no non-trivial analytic subvarieties, then Proof. Note that for every β ∈ B, by [16, page 167, Corollary 7.4] (our definition of dd c in Definition 1.5 is half of the one there) we have
On the other hand, we have the following identity proved by Tosatti in [60, Theorem 4.6] where the infimum runs over all positive-dimensional irreducible analytic subvarieties V containing x and mult x V denotes the multiplicity of V at x. Hence if X has no non-trivial subvarieties then (put β 0 = (1/n, · · · , 1/n), use (3.1) and the remark above)

Relation with the s-invariant.
Our β-Seshadri constant is closely related to the s-invariant introduced by Cutkosky, Ein and Lazarsfeld in [14].
Theorem 3.2. Let (X, ω) be an n-dimensional compact Kähler manifold. Assume that ω lies in the first Chern class of a holomorphic line bundle L on X. Fix β = (β 1 , · · · , β n ) ∈ R n such that all β −1 j are positive integers. Then where I β is the ideal of O X generated by {z Proof. First let us prove ǫ x (ω; β) ≥ 1 s L (I β ) . Note that L is ample since ω is positive, hence 1 s L (I β ) = sup{γ ≥ 0 : µ * L − γE is ample}.
By Example 5.4.10 in [41], one may replace µ by a desingularization f : Y → X of I β with exceptional divisor F , more precisely, we have which implies that for every γ < 1 s L (I β ) there is a singular metric e −φ on f * L with γ-log pole along F such that i∂∂φ > 0 on Y . Then the weight f * φ on L will have the γT β -singularity, from which we know that ǫ x (ω; β) ≥ γ. Hence ǫ x (ω; β) ≥ 1 s L (I β ) . Now let us prove that ǫ x (ω; β) ≤ 1 s L (I β ) . For every γ < ǫ x (ω; β), we can find a singular metric e −ψ on L with γT β -singularity such that i∂∂ψ > 0. Then e −f * ψ defines a singular metric on f * L with γ-log pole along F such that i∂∂(f * ψ) > 0 on Y , from which we know that ǫ x (ω; β) ≤ 1 s L (I β ) . Remark. Since I β are special monomial ideals, it is not hard to find the explicit desingularizations. Let us look at the simple example β = (1, 1 2 ). Then, in this case, we have I β = Span{z 1 , z 2 2 }. First, one may blow up the origin, so z 1 = uv, z 2 = u gives Bl 0 I β = Span{uv, u 2 }, then we can blow up the point (u, v) = (0, 0) :=0, so v = ts, u = s gives Bl0Bl 0 I β = Span{s 2 }, from which we know that F = 2 · |{s = 0}|.

Extremal property of the β-Seshadri constant. For s-invariant of a general monomial ideal
I P := Span{z α 1 , · · · , z α k }, α j ∈ Z n >0 , z α j := z α j 1 1 · · · z α j n n , with isolated zero set {x}, where P is the Newton polytope defined by P := convex hull of ∪ 1≤j≤k P α j , P α j := {x ∈ R n : x l ≥ α j l , 1 ≤ l ≤ n}, one may correspondingly define the P -Seshadri constant |z α | 2 , v(P ) denotes the set of vertices of P .
Then the proof of Theorem 3.2 also implies .
The reason why we only use β-Seshadri constants in this paper is that they have the following extremal property.
Remark. The condition that (1, · · · , 1) lies in the boundary of P is equivalent to the following identity sup{c ≥ 0 : e −cT P is integrable near z = 0} = 1, see [33,31] for the proof and related results.
In order to generalize the above proof to general β-Seshadri constants, we need to construct the associated β-version of ψ r in (3.8) (see Lemma 3.8 below). In the next subsection, we shall use the Legendre transform theory to decode the construction.  the Legendre transform of φ. Let A ⊂ R n be a closed set. We call the iterated Legendre transform of φ with respect to A.

Remark. Notice that
Lemma 3.5. Let φ be a smooth convex function on R n . Then Proof. It follows from the fact that x is the maximum point of the following concave function if and only if x is a critical point of ψ α .
Proposition 3.6. If φ is smooth strictly convex and A ⊂ R n is closed then Proof. By the above lemma, we have with identity holds if and only if α = ∇φ(x). Hence ∇φ(x) is the unique maximum point of the following function ρ x : α → α · x − φ * (α). The proof of Proposition 2.2 in [62] implies that ρ x is smooth strictly concave on ∇φ(R n ). Hence the supremum of ρ x on the complement of any small ball around ∇φ(x) must be strictly smaller than φ(x). By our assumption, A is closed, thus φ(x) = φ A (x) if and only if ∇φ(x) ∈ A.
Definition 3.2. Let φ be a smooth strictly convex function on R n and A be a closed set in R n . We call The above proposition implies that (3.10) Our key observation is the following: where ∂A denotes the boundary of A.
Proof. Since ∇φ(x) is the unique maximum point of the following concave function and ∇φ(x) / ∈ A, we know that for every point a ∈ A, is increasing on t ∈ [0, 1]. Taket ∈ [0, 1] such that Hence the theorem follows.
Proof of Theorem A. Similar as the proof of Theorem 3.4 (replace B r by B β r ), we know that the lemma above gives Theorem A.

3.6.
A partial converse of the Hörmander criterion. For general higher dimensional cases, we do not know whether the Hörmander criterion is an equivalent criterion or not. Based on Demailly-Pȃun's generalized Nakai-Moishezon ampleness criterion [19], a theorem of Nakamaye [48,51], Lindholm's result [42] and the Balian-Low type theorem (see [32,Theorem 10] and [1, Theorem 1.5]), we obtain the following partial converse of the Hörmander criterion. Theorem 3.9. Let Γ be a lattice in C n . Denote by ι Γ the Hörmander constant of (C n /Γ, ω euc ).
(1) If Γ is a set of interpolation for F 2 and all irreducible analytic subvarieties of X are translates of complex tori then ι Γ > 1/n; (2) If Γ is a set of interpolation for F 2 and the only positive dimensional irreducible analytic subvariety of X is X itself then ι Γ = (n! |Γ|) 1/n n > (n!) 1/n n ; (3) Assume that ω euc is rational on Γ or the Picard number of X is n 2 . If Γ is a set of interpolation for F 2 then ι Γ > 1/(n e).
Proof. Denote by 0 the unit of the torus and write ω := ω euc . The main idea is to use the following Demailly-Pȃun identity proved by Tosatti in [60, Theorem 4.6] where the infimum runs over all positive-dimensional irreducible analytic subvarieties V containing 0, and mult 0 V denotes the multiplicity of V at 0. Now let us prove Theorem (1), by our assumption, it suffices to show that for all complex subtorus Choose a C linear subspace E of C n such that V = E/(E ∩ Γ). Notice that if Γ is a set of interpolation for F 2 , then E ∩ Γ is a set of interpolation for F 2 | E . Thus the Balian-Low type theorem implies that which gives ǫ 0 (ω) > 1, hence (1) follows. Now assume further that X has no non-trivial subvarieties, then (3.2) implies (nι Γ ) n /n! = |X| = |Γ|, thus (2) follows directly from the Balian-Low type theorem. To prove (3), we shall use the following inequality (see [35,Lemma 3 where the infimum runs over all positive-dimensional abelian subvarieties V containing 0. Notice that the right hand side of (3.12) is 1-homogeneous with respect to ω, we know that (3.12) also holds for all ω = cω ′ , where ω ′ is integral on Γ. In particular, it holds true if ω is rational on Γ.
In case the Picard number of X is n 2 , we know that ω can be approximated by rational ω ′ , hence (3.12) is true for all ω. By Balian-Low type theorem, we have By Stirling's approximation, we have hence (3) follows.

3.7.
Hörmander constants and densities of general discrete sets. Definition 3.3. Let S be a discrete set in C n . Let ψ be a non positive function such that ψ + π|z| 2 is plurisubharmonic on C n . Let γ be positive number. We call (ψ, γ) an S-admissible pair if ψ is smooth outside S, e −ψ/γ is not integrable near every point in S and there exists a small constant ε 0 > 0 such that Assume that there exists an S-admissible pair, then we call ι(S) := sup{γ > 0 : there exists ψ such that (ψ, γ) is S-admissible} the Hörmander constant of S.
Since ψ equals to −∞ at S, (3.13) implies that thus S is uniformly discrete. The proof of the Hörmander criterion also implies: Theorem 3.10. Let S be a discrete set in C n . Assume that ι(S) > 1. Then there exists a constant C > 0 such that for every sequence of complex numbers a = {a λ } λ∈S with there exists F ∈ F 2 such that F (λ) = a λ for all λ ∈ S and ||F || 2 ≤ C. In case S is a lattice in C n , we know that D + (S) −1 is equal to the Lebesgue measure of the torus C n /S. In the one-dimensional case, we also have the following general result.
Using results from [8,52], one may also generalize the above theorem to general weight function φ with φ zz bounded by two positive constants. For general higher-dimensional cases, by [42,Theorem 2], we know that if S is F 2 interpolating then D + (S) ≤ 1. However, in general, S may not be F 2 interpolating even D + (S) is small enough. Comparing with Theorem 3.10, this means that there exists S with very small upper uniform density whose Hörmander constant is also small.
gives a frame in L 2 (R 2 ) then (Z 2 , e −πt 2 ) defines a frame in L 2 (R), which is not true by the Balian-Low type theorem. Now it suffices to prove (2) since (2) implies (1) by the remark after Corollary 1.9. Put then we know that (X, ω) is of type (1,4). Since the moduli space of polarized type (d 1 , d 2 ) (d 1 , d 2 are fixed positive integers) Abelian surfaces is equal to the Siegel upper half-space, to prove (2), by the Hörmander criterion, it suffices to show that the Seshadri constant of a generic polarized type (1, 4) Abelian surface is bigger than two. Since generically a polarized type (1,4) Abelian surface has Picard number one, by Theorem 6.1 (b) in [2], its Seshadri constant equals where we use the fact that k = 1, l = 3 is the primitive solution of the following Pell's equation The proof is complete.

NON TRANSCENDENTAL EXAMPLES
4.1. Gröchenig-Lyubarskii's example. Let us look at the following lattice in R 2 ×R 2 (see [27, page 3, (4)]): Its symplectic dual is Fix Ω = iI n , where I n denotes the identity matrix. With the notation in Proposition 1.4, we have Let us estimate the Seshadri constant of dz j ∧ dz j on the complex tori X := C 2 /Γ Ω,Λ • . Notice that the Riemannian metric induced by ω is precisely the euclidean metric | · |, hence Thus the following ball B := {|z| < 1/ √ 3} contains precisely one point in Γ Ω,Λ • and we can think of B as a Kähler ball in X, which gives (see Theorem 3.4) the following Seshadri constant inequality However, from [27, page 3, (4)], we know that (Λ, g Γ ) does not define a frame in L 2 (R 2 ). Thus by Proposition 1.4, Γ Ω,Λ • is not a set of interpolation for F 2 . To summarize, we obtain: There exists a lattice in C 2 whose Seshadri constant is bigger than one but it is not a set of interpolation for F 2 .
Remark. By the Hörmander criterion, we know every lattice in C 2 with Seshadri constant bigger than two is a set of interpolation for F 2 . In the above example, one may further prove that In fact the complex line C(0, 2/ √ 3) covers a subtorus, say of X, thus [60,Theorem 4.6] implies that from which (4.2) follows.

Complex lattices.
We call Γ a complex lattice if iΓ = Γ.
One may verify the following: For a lattice Γ in C n , the followings are equivalent (1) Γ is a complex lattice; (2) Γ = Z[i]{γ 1 , · · · , γ n } for some γ j ∈ C n ; (3) Γ = AZ[i] n for some A ∈ GL(n, C); Now we can prove a generalization of (4.2). Proof. Choose z = A −1 w as the new variable, one may assume that A = I n . Then Denote by ψ 0 (z) the Green function on C/Z[i] satisfying iπdz ∧ dz + i∂∂ψ 0 = i∂∂ log |z| 2 .
then we know that ψ := e min (A) · max{ψ 0 (z 1 ), · · · , ψ 0 (z n )} satisfies 2πω + i∂∂ψ ≥ 0 and has order one log pole at Γ. Thus we know that ǫ 0 (ω) ≥ e min (A), together with Theorem 3.4, it gives the lower bound of the Seshadri constant that we need. To prove the upper bound, it suffices to choose a subtorus then [60,Theorem 4.6] gives Take the infimum over all 0 = γ ∈ Γ, the upper bound follows.

Seshadri sequence and Gröchenig's result.
In this section, we shall rephrase the main result of Gröchenig in [24] in terms of the Seshadri constant. The main idea is to consider a sequence of extensions, more precisely, let be an increasing sequence of complex Lie subgroups of X. We shall introduce the Seshadri constant ǫ j , 1 ≤ j ≤ k, for extension from X j−1 to X j . Let π j : E j → X j be the covering map, where E j is an n j dimensional complex subspace of C n . Let be the orthogonal decomposition with respect to the Euclidean metric ω. Then is not a subtorus of X j ). Denote by (4.4) ǫ j := ǫ 0 (ω; X ⊥ j−1 ) the Seshadri constant at the origin of X ⊥ j−1 with respect to ω. Definition 4.1. We call (4.3) an admissible sequence of X if ǫ j > n j − n j−1 , ∀ 1 ≤ j ≤ n.
X is said to be Seshadri admissible if it possesses an admissible sequence.
Theorem 4.4. Assume that X is Seshadri admissible, then Γ is a set of interpolation in F 2 .
Proof. Let π : C n → X be the covering map. Put F 2 j := F holomorphic on π −1 (X j ) : It suffices to prove that each element f in F 2 j−1 extends to an element F in F 2 j with ||F || ≤ C j ||f ||. Since π −1 (X j ) is a disjoint union of translates of E j and π −1 (X j−1 ) ∩ E j = π −1 j (X j−1 ), the above extension problem reduces to the extension from F 2 j−1 | π −1 j (X j−1 ) to F 2 j | E j . Apply the Hörmander method, it suffices to construct an ω plurisubharmonic function with order (n j −n j−1 ) log pole along π −1 j (X j−1 ). Now the assumption ǫ j > n j − n j−1 gives an ω plurisubharmonic ψ j with order n j − n j−1 log pole at the origin of F j , the pull back of ψ j along the natural projection E j → F j gives the function that we need. Now we shall show how to use the above theorem to give a new proof of [24, Theorem 9] on the Gabor frame property for (Λ, g Ω ). The setup for [24,Theorem 9] is the following: Based on Proposition 1.4, we shall prove a similar result with a weaker assumption, i.e. we shall only assume that Γ Ω,Λ • is a complex lattice. Let us write where A ∈ GL(n, C). By the Iwasawa decomposition (see [11,Proposition 26.1]), we have where U is unitary and S is lower triangular with positive eigenvalues λ j (U, S are uniquely determined by A, λ −1 j is equal to γ j in [24,Theorem 9]). Since the Euclidean metric ω is unitary invariant, one may assume that Then we have F j ≃ C, Γ j ≃ λ n−j+1 Z[i], and ǫ j = λ 2 n−j+1 , 1 ≤ j ≤ n. Since n j = j, we have n j − n j−1 = 1. Hence if λ n−j+1 > 1, 1 ≤ j ≤ n, then X is Seshadri admissible. Thus Theorem 4.4 implies the following slight generalization of Gröchenig's result (notice again that γ j = λ −1 j ): Theorem 4.5 (Theorem 9 in [24]). Let Γ Ω,Λ • be a complex lattice. With the notation above, assume that λ j > 1 for all 1 ≤ j ≤ n. Then Γ Ω,Λ • is set of interpolation for F 2 (and equivalently (Λ, g Ω ) defines a frame in L 2 (R n )).
Proof of the lower bound. Notice that (by an induction on n) Notice that The main observation is that the Taylor expansion is now an orthogonal decomposition, i.e.
from which we know that Now put t = π|γ| 2 , we have which gives (5.2).
Proof of the upper bound. The main idea is to use Theorem 2.9. The definition of C implies that B := {z ∈ C n : π|z| 2 < C} is embedded ball in X := C n /Γ. For π C ≤ δ < π n , Then we have i∂∂ψ δ ≥ −δ · i∂∂|z| 2 . Denote by ψ the pull back to C n of πψ δ /δ. Apply Theorem 2.9 to ψ, we get then n/C ≤ x < 1 and Thus the upper bound follows from inf n/C≤x<1 n n (1 − x)x n = M(C). The proof of Theorem B is now complete.

5.2.
Interpolation bounds in terms of the Robin constant. In this subsection, we shall generalize Theorem 5.1 to the case that ǫ 0 (ω) > n. The main idea is to consider the following envelope with prescribed singularity ψ π a := sup{ψ 0 ≤ 0 : 2πω + i∂∂ψ 0 ≥ 0, ψ 0 has an isolated order a log pole at the origin} on the torus X = C n /Γ. Denote by ψ a the pull back to C n of ψ π a . Definition 5.1. We call ψ a the a-envelope function on C n associated to the lattice Γ and ρ a := lim inf z→0 ψ a (z) − a log |z| 2 .
the a-Robin constant of Γ.
We have the following generalization of Theorem 5.1 and Theorem B.
Theorem 5.2. Assume that ǫ := ǫ 0 (ω) > n. Put Then every sequence of complex numbers c = {c γ } with γ∈Γ |c γ | 2 e −π|γ| 2 = 1 extends to a function in F 2 . Moreover, Proof. The proof of the lower bound is the same. For the upper bound, it suffices to apply Theorem 2.9 to ψ = ψ a .