Balian-Low type theorems in finite dimensions

We formulate and prove finite dimensional analogs for the classical Balian-Low theorem, and for a quantitative Balian-Low type theorem that, in the case of the real line, we obtained in a previous work. Moreover, we show that these results imply their counter-parts on the real line.

1. Introduction 1.1. A Balian-Low type theorem in finite dimensions. Let g ∈ L 2 (R). The Gabor system generated by g with respect to the lattice Z 2 is given by The classical Balian-Low theorem [3,4,9,21] states that if the Gabor system G(g) is an orthonormal basis, or a Riesz basis, in L 2 (R), then g must have much worse time-frequency localization than what the uncertainty principle permits. The precise formulation is as follows (see [7] for a detailed discussion of the proof and its history).
Theorem A (Balian, Battle, Coifman, Daubechies, Low, Semmes). Let g ∈ L 2 (R). If G(g) is an orthonormal basis or a Riesz basis in L 2 (R), then R |t| 2 |g(t)| 2 dt = ∞ or R |ξ| 2 |ĝ(ξ)| 2 dξ = ∞. (2) We note that by Parseval's identity, condition (2) is equivalent to saying that we must have either R |ĝ ′ (ξ)| 2 dξ = ∞ or R |g ′ (t)| 2 dt = ∞. ( That is, these integrals are considered infinite if the corresponding functions are not absolutely continuous, or if they do not have a derivative in L 2 . In the last 25 years, the Balian-Low theorem inspired a large body of work in time-frequency analysis, including, among others, a non-symmetric version [6,12,13,23], an amalagam space version [19], versions which discuss different types of systems [10,17,18,23], versions not on lattices [8,16], and a quantified version [24]. The latter result will be discussed in more detail in the second part of this introduction.
Although it provides for an excellent "rule of thumbs" in time-frequency analysis, the Balian-Low theorem is not adaptable to many applications since, in realistic situations, information about a signal is given by a finite dimensional vector rather than by a function over the real line. The question of whether a finite dimensional version of this theorem holds has been circling among researchers in the area 1 . In particular, Lammers and Stampe pose this as the "finite dimensional Balian-Low conjecture" in [20]. Our main goal in this paper is to answer this question in the affirmative.
Let N ∈ N and denote d = N 2 . We consider the space ℓ d 2 of all functions defined over the cyclic group Z d := Z/dZ with the normalization, To motivate this normalization, let g be a continuous function in L 2 (R) and put b(j) = g(j/N), j ∈ Z ∩ [−N 2 /2, N 2 /2). That is, the sequence b ∈ ℓ d 2 consists of samples of the function g, at steps of length 1/N over the interval [−N/2, N/2]. Then, for "large enough" N, the above ℓ 2 norm can be interpreted as a Riemann sum approximating the L 2 (R) norm of g. Note that in Section 2, we define the finite Fourier transform F d so that it is unitary on ℓ d 2 . Let NZ d denote the set {Nk : k ∈ Z} modulo d. For b ∈ ℓ d 2 , the Gabor system generated by b with respect to (NZ d ) 2 , is given by We point out that, with the choice b(j) = g(j/N), the discrete Gabor system G d (b) yields a discretization of the Gabor system G(g) restricted to the interval [−N/2, N/2).
To formulate the Balian-Low theorem in this setting, we use a discrete version of condition (3). To this end, we denote the discrete derivative of a function b = {b(j)} d−1 j=0 ∈ ℓ d 2 by ∆b : , and put where the infimum is taken over all sequences b ∈ ℓ d 2 for which the system G d (b) is an orthonormal basis in ℓ d 2 . We note that for the choice b(j) = g(j/N), samples of the derivative of g at the points j/N are approximated by N∆b. Therefore, the expression inside of the infimum is a discretization of the integrals in the condition (3). Our finite dimensional version of the Balian-Low theorem, that answers the finite Balian-Low conjecture in the affirmitive, may now be formulated as follows.
Theorem 1.1. There exist constants c, C > 0 so that, for all integers N ≥ 2, we have c log N ≤ α(N) ≤ C log N.
In particular, α(N) → ∞ as N tends to infinity.
Remark 1.2. Theorem 1.1 also holds in the case that the infimum in α(N) is taken over all b ∈ ℓ d 2 for which the system G d (b) is a basis in ℓ d 2 with lower and upper Riesz basis bounds at least A and at most B, respectively. In this case, the constants c, C in Theorem 1.1 depend on A and B. (For a precise definition of the Riesz basis bounds see Section 2). The dependence on the Riesz basis bounds is necessary, in the sense that it can not be replaced by a dependence on the ℓ d 1.2. A finite dimensional quantitative Balian-Low type theorem. In [22], F. Nazarov obtained the following quantitative version of the classical uncertainty principle: Let g ∈ L 2 (R) and Q, R ⊂ R be two sets of finite measure, then R\Q |g(t)| 2 dt + R\R |ĝ(ξ)| 2 dξ ≥ De −C|Q||R| g 2 L 2 (R) .
In [24], we obtained the following quantitative Balian-Low theorem, which is a modest analog of Nazarov's result for generators of Gabor orthonormal bases and, more generally, Gabor Riesz bases.
Theorem B (Nitzan, Olsen). Let g ∈ L 2 (R). If G(g) is an orthonormal basis or a Riesz basis then, for every Q, R > 1, we have where the constant C > 0 depends only on the Riesz basis bounds of G(g).
This quantitative version of the Balian-Low theorem implies the classical Balian-Low theorem (Theorem A), as well as several extensions of it, including the nonsymmetric cases and the amalgam space cases referred to above. Here, we prove the following finite dimensional version of this theorem. Theorem 1.5. There exists a constant C > 0 such that the following holds. Let Remark 1.6. Theorem 1.5 holds for general bases as well. In this case, the constant C, as well as the conditions on the sizes of N, Q and R, depend on the Riesz basis bounds. This more general version of Theorem 1.5 is formulated as Theorem 5.3 in Section 4. (For a precise definition of the Riesz basis bounds see Section 2). Remark 1.7. As we show in Section 6, the quantitative Balian-Low theorem (Theorem B) follows as a corollary of Theorem 1.5.
Remark 1.8. The conditions appearing in Theorem 1.5 are not optimal, but rather, these conditions were chosen to avoid a cumbersome presentation. In particular, we point out that a more delicate estimate in Lemma 5.1, or the use of a different function, will improve the condition N ≥ 200. Some modifications in the proof of Lemma 5.2 will improve this condition as well. In addition, a careful analysis of the proof will allow one to improve each one of the conditions N ≥ 200 and Q, R ≤ N/16 at the expense of making the constant C smaller. In fact, any two of the previous conditions can be improved at the cost of the third.
1.3. Finite dimension Balian-Low type theorems over rectangular lattices. The conclusions of the classical Balian-Low theorem (Theorem A) and its quantitative version (Theorem B), still hold if we replace Gabor systems over the square lattice Z 2 by Gabor systems over the rectangular lattices λZ × 1 λ Z, where λ > 0. Indeed, this is immediately seen by making an appropriate dilation of the generator function g. In the finite dimensional case, however, such dilations are in general not possible. The question of which finite rectangular lattices allow Balian-Low type theorems therefore has an interest in its own right. We address this in the extensions of of theorems 1.1 and 1.5 formulated below.
Let M, N ∈ N and denote d = MN. We consider the space ℓ (M,N ) 2 of all functions defined over the cyclic group This non-symmetric normalization is motivated by the fact that if g is a continuous function in L 2 (R) and b(j) = g(j/M), j ∈ Z ∩ [−MN/2, MN/2), then the above ℓ 2 -norm can be interpreted as a Riemann sum for the L 2 (R) norm of g over the interval [−N/2, N/2]. Note that in Section 7, we define the finite Fourier transform F (M,N ) so that it is a unitary operator from ℓ . The Gabor system generated by b with respect to MZ d × NZ d is given by We point out that making the choice b(j) = g(j/M), the discrete Gabor system G (M,N ) (b) yields a 1/M-discretization of the Gabor system G(g) restricted to To formulate the discrete Balian-Low theorem in this setting, we put where the infimum is taken over all sequences b ∈ ℓ (M,N ) 2 for which the system . We note that for the choice b(j) = g(j/M), the expression inside of the infimum is a discretization of the integrals in condition (3).
We are now ready to formulate the extension of Theorem 1.1 to Gabor systems over rectangles. Theorem 1.9. There exist constants c, C > 0 so that, for all integers M, N ≥ 2, we have c log min{M, N} ≤ α(M, N) ≤ C log min{M, N}. In particular, α(M, N) → ∞ as min{M, N} tends to infinity.
Similarly, the corresponding extension of Theorem 1.5 may be formulated as follows.
We point out that remarks 1.2 and 1.6 also hold for theorems 1.9 and 1.10, respectively. That is, these theorems can be extended to generators of general bases with the constants depending only on the Riesz basis bounds. Remark 1.8 also holds in this case.
1.4. The structure of the paper. In Section 2, we discuss some preliminaries, in particular the finite and continuous Zak transform and their use in characterizing orthonormal Gabor bases and Riesz Gabor bases. In Section 3, we present two improved versions of a lemma we first proved in [24]. These results quantify the discontinuity of the argument of a quasi-periodic function. In Section 4, we apply these lemmas to prove Theorem 1.1, while in Section 5, we use them to prove Theorem 1.5. In Section 6, we show how the Balian-Low theorem (Theorem A) and its quantitative version (Theorem B) can be obtained from their finite dimensional analogs. Finally, in Section 7, we discuss theorems 1.9 and 1.10. In the most part, we give only a sketch of the proofs for the rectangular lattice case, as they are very similar to the proofs we present for the square lattice case.

Preliminaries
2.1. Basic notations, and the continuous and finite Fourier transforms. Throughout the paper, we usually denote by f a function defined over the real line, and by g a function defined over the real line which is a generator of a Gabor system. Similarly, we usually denote by a a discrete function in ℓ d 2 , and by b a function in ℓ d 2 which is a generator of a Gabor system. For f ∈ L 1 (R), and with the usual extension to f ∈ L 2 (R), we use the Fourier transform We let S(R) denote the Schwartz class of functions φ which are infinitely many times differentiable, and that satisfy sup t∈R |t k φ (ℓ) (t)| < ∞ for all k, ℓ ∈ N.
Recall from the introduction that for N ∈ N and d = N 2 , we denote by ℓ d 2 the space of all functions defined over the cyclic group Z d := Z/dZ with the normalization For a ∈ ℓ d 2 , we use the finite Fourier transform With the chosen normalization, the finite Fourier transform is unitary on ℓ d 2 . We define the periodic convolution of a, b ∈ C d , by and note the convolution relation F d (a * b) = F d (a) · F d (b). Observe that, for the choice a(j) = f (j/N), the discrete Fourier transforms and convolutions yield natural discretizations of their respective counterparts on R. Also, recall that for a sequence a ∈ ℓ d 2 , we denote the discrete derivative by ∆a(j) = a(j + 1) − a(j). From time to time, we encounter sequences depending on more than one variable, say a k+ψ(s) where ψ is some function depending on the integer s. In this case, we write ∆ (s) if we want to indicate that the difference is to be taken with respect to s. That is, ∆ (s) a k+ψ(s) = a k+ψ(s+1) −a k+ψ(s) .
where, as usual, d = N 2 . We note that this normalization can be related to the process of sampling an L 2 function over [0, 1] 2 , on the vertices of squares of side length 1/N, and computing the corresponding Rieman sum.
2.2. The continuous and finite Zak transforms. On L 2 (R), the Zak transform is defined as follows.
Definition 2.1. Let f ∈ L 2 (R). The continuous Zak transform of f is given by We summarise the basic properties of the continuous Zak transform in the following lemma. Proofs for these properties, as well as further discussion of the Zak transform, can be found, e.g., in [14, Chapter 8].
In particular, this means that the function Zf is determined by its values on (iv) For φ ∈ S(R), the Zak transform satisfies the convolution relation where the subscript of * 1 indicates that the convolution is taken with respect to the first variable of the Zak transform.
Definition 2.3. Let N ∈ Z and set d = N 2 . The finite Zak transform of a ∈ ℓ d 2 , with respect to (NZ d ) 2 , is given by Note that with this definition, Z d (a) is well-defined as a function on Z 2 d (that is, it is d-periodic separately in each variable).
The basic properties of the finite Zak transform mirror closely those of the continuous Zak transform and are stated in the following lemma. Parts (i), (ii) and (iii) of this Lemma can be found as theorems 1, 3 and 4 in [2]. Part (iv) follows immediately from the definitions of the Zak transform and the convolution.
Then the following hold.
In particular, Z d (a) is determined by its values on the set (iv) The finite Zak transform satisfies the convolution relation where the subscript of * 1 indicates that the convolution is taken with respect to the first variable of the finite Zak transform.
Remark 2.5. We will make use of a somewhat more general property than the N-quasi periodicity. Namely, we will be interested in functions W : where η is a unimodular constant. In particular, we note that if a function is N-quasi-periodic then any translation of it satisfies the relations (14). For easy reference to this property we will call a function satisfying it N-quasi-periodic up to a constant.
We will make use of the following lemma, which is a finite dimensional analog of inequality (16) from [24].
Lemma 2.6. Let N ∈ N and d = N 2 . Suppose that a, φ ∈ ℓ d 2 and k ∈ N, then it holds that for all finite sequences of complex numbers {c n }, where A and B are positive constants. The largest A and smallest B for which (16) holds are called the lower and upper Riesz basis bounds, respectively. We note that every basis in a finite dimensional space is a Riesz basis. The proof for Part (i) of the following proposition can be found, e.g., in [14, Proposition 2.7. (i) Let g ∈ L 2 (R). Then, G(g) is a Riesz basis in L 2 (R) with

Riesz basis bounds A and B if and only if
2.4. Relating continuous and finite signals. In the introduction, we motivated our choices of normalizations by relating finite signals to samples of continuous ones. In this subsection, we formulate this relation precisely.
Fix N ∈ N and let d = N 2 . For a function f in the Schwartz class S(R), we denote its N-periodisation by and the N-samples of a continuous N-periodic function h by .
First we relate these operators to the Fourier transform and the Zak transform via Poisson-type formulas. We note that part (i) of the following proposition is stated without proof in [1].
Proposition 2.8. For f in the Schwartz class S(R), the following hold.
(i) For every N ∈ N and (m, n) ∈ Z 2 , we have (ii) For every N ∈ N, we have (ii): Observe that part (i) holds for both f and F f . With this, in combination with parts (iii) of Lemma 2.2 and Lemma 2.4, the proof of part (ii) follows.
Remark 2.9. (i) Although Proposition 2.8 is formulated for functions in the Schwartz class S(R), it is readily checked that it holds for all functions f ∈ L 2 (R) which satisfy both sup t∈R |t 2 f (t)| < ∞ and sup ξ∈R |ξ 2f (ξ)| < ∞. (ii) In Section 6, we obtain a version of Proposition 2.8 which holds for all f ∈ L 2 (R).
We end this section with a lemma that relates the discrete and continuous derivatives.
Lemma 2.10. Let f ∈ L 2 (R) be a function that satisfies the condition of Remark 2.9(i), and denote a = S N P N f . Then,

Regularity of the Zak transforms
Essentially, this paper is about the regularity of Zak transforms (or rather, their lack of such). In this section, we formulate a few lemmas in this regard.
3.1. 'Jumps' of quasi-periodic functions on Z 2 d . It is well known that the argument of a quasi-periodic function on R 2 cannot be continuous (see, e.g., [14, Lemma 8.4.2]). In [24, Lemma 1], we show that such a function has to 'jump' on all rectangular lattices (see Remark 3.2, below). The latter lemma is finite dimensional in nature. Below, we formulate it as such, as well as improve the constants used in the lemma. To this end, we use the notation or Remark 3.2. From time to time, we refer to functions satisfying conditions such as (19) or (20) as having 'jumps'. This notion of 'jumps' is imprecise and merely meant to be descriptive, and will depend on the context of the given situation. Proof of Lemma 3.1. To obtain a contradiction, we assume that neither (19) nor (20) hold. For the sake of convenience, we choose a specific branch of the function H as follows: The conditions on H imply that for all integers 0 ≤ i ≤ K and 1 ≤ j ≤ L, there exist integers M i and N j so that In particular, plugging j = 0, j = L and i = 0, i = K into these equations, it is easy to check that We will now show that our choice of branch implies on the one hand, that N L = N 0 , and, on the other hand, that M K = M 0 . This contradicts (22) and would therefore complete the proof.
To show that N L = N 0 , we note that due to the relations (21), we have which, by the triangle inequality, together with the conditions L ≥ 2 and |Γh i, Since all the N j are integers, this implies that all N j are identical and, in particular, that N L = N 0 . On the other hand, to show that Such n i,j and α i,j exist due to our assumption that (19) does not hold. Then, As above, the triangle inequality combined with our assumptions imply that n i,j = n i,j+1 for every i, j. In particular, we have , and since all the M i are integers, we conclude that M i = M i+1 for every i. This completes the proof.
3.2. 'Jumps' of quasi-periodic functions on subsets of Z 2 d . In this subsection, we extend Lemma 3.1 to show that it also holds when the function is restricted to certain subsets of Z 2 d , which we want to treat as if they were sublattices of Z 2 d , even if they, strictly speaking, are not. To this end, for integers K, L ∈ [2, N], we define the functions where [a] denotes the integer part of a. Note that σ K = ω L = N. We can now state the following lemma.
, and denote d = N 2 . Let W be a function defined over Z 2 d that is N-quasi-periodic up to a constant (see (14)). Denote by H any branch of the argument of W , so that W = |W |e 2πiH . or We note that since ω L = N, and H is the argument of a function that is N-periodic in the second variable, we have hold. The jumps for h s,t now follow from the jumps of Φ. Indeed, if the jump is in the vertical direction, or if it is in the horizontal direction at a point (s, t) with s ≤ K − 2, then this is immediate from the definition of Φ. If the jump is in the horizontal direction, at a point (s, t) with s = K − 1, then we note that, since σ K = N, the N-quasi-periodicity up to a constant of W implies that The lemma now follows from the fact that, We obtain the following corollary of Lemma 3.4.
Corollary 3.6. Fix an integer N 0 ≥ 5 and a constant A > 0. Let For any integers N ≥ N 0 and K, L ∈ [2, N], the following holds where σ s , ω t are defined in (23).
Proof. First, we point out that we define the argument arg(z) of a complex number z so that z = |z|e 2πiarg(z) . Since any translation of a quasi-periodic function is quasi-periodic up to a constant, as defined in (14), it follows from Lemma 3.4 that on an N × N square the argument of W jumps by more than 1/4 − 1/N in at least one of the inequalities (26) or (27). As the modulus of W is bounded from below by √ A, the conclusion now follows from basic trigonometry.

4.
A proof for Theorem 1.1 Here we give a proof for Theorem 1.1 in the general Riesz basis case referred to in Remark 1.2. In the first subsection below, we reformulate the theorem in terms of the Zak transform. Using this, we proceed to prove the bound from below, and, finally, we prove the bound from above.

4.1.
Measures of smoothness for finite sequences. Fix N ∈ N and set d = Note that with these notations the quantity α(N) defined in the introduction satisfies where the infimum is taken over b ∈ ℓ d 2 for which G(b) is an orthonormal basis.
Proof. We will only prove the right-hand side inequality in (28) since the left-hand side inequality is proved in the same way.
As the finite Zak transform commutes with the difference operation ∆, that is Z d (∆b) = ∆Z d (b), and the finite Zak transform is unitary from ℓ d 2 to ℓ 2 ([0, N −1] 2 ), we find that Similarly, To relate the expression on the right-hand side to ΓZ d (b), we use the relation between the finite Fourier transform and the Zak transform (13) to compute Combining this estimate with (30), and recalling that the Zak transform is Nperiodic in the second variable, we find that where, in the last estimate, we used the facts that m ≤ N and d = N 2 . Multiplying this inequality by N 2 , and combining it with (29), the right-hand inequality of (28) follows.
Next, for A, B > 0, we put where the infimums are taken over all b ∈ ℓ d 2 for which the system G d (b) is a basis with Riesz basis bounds at least A and at most B. Proposition 4.1 now implies that with the notations above, the following inequality holds for every N ∈ N: In light of this inequality, Theorem 1.1 (as well the version discussed in Remark 1.2) can be reformulated as follows.  To see that it is necessary to include the the Riesz basis bounds in the definitions of α A,B (N) and β A,B (N) (and that these bounds cannot be replaced by ℓ d 2 normalization) consider the following example. Let h = e −π(x−τ ) 2 with τ ∈ (0, 1/2)\Q. By [11,Lemma 3.40], it follows that Zh has exactly one zero on the unit square located at (1/2 + τ, 1/2). Since the first coordinate of this point is irrational, it follows by Proposition 2.8 that the function Consequently, Proposition 2.7 implies that the finite Gabor system G d (b N ) is a basis for ℓ d 2 , though the (lower) Riesz basis bounds of these bases decay as N increases. Straightforward computations, using only the regularity and decay of the Gaussian, show that there exist constants C, D, E > 0, such that the following hold: Proof. Given N ≥ 5, we set d = N 2 and let J ∈ N be such that t be as defined in (23) We note that where both the infimum and supremum are taken over all t ∈ Z. Indeed, to see this, consider separately the cases N = 2 J and N > 2 J and use the fact that all of the numbers involved in these inequalities are integers.
A sin(π/20), then, since N ≥ 5 Corollary 3.6 implies that each of the sets Lat j (u, v) contains a point on which the function W 'jumps', i.e., where Our goal is to collect 'jumps' of W that are, in some sense, separated. We do this in an inductive process.
In the first step, let j = 0. By Corollary 3.6, there exists a point (m 0 , n 0 ) in Lat 0 (0, 0) so that (33) holds for this point. Let S 0 = S 0 = {(m 0 , n 0 )}. Next, let j = 1. For u ∈ {0, 1}, the sets Lat 1 (u) are disjoint, and so at least one of them does not contain the number m 0 . Let u 1 1 ∈ {0, 1} be such that the set Lat 1 (u 1 1 ) has this property, and, similarly, let v 1 1 ∈ {0, 1} be such that Lat 1 (v 1 1 ) does not contain the number n 0 . By Corollary 3.6, there exists a point (m 1 , n 1 ) in Lat 1 (u 1 1 , v 1 1 ) so that (33) holds for this point. Let S 1 = {(m 1 , n 1 )}, and put S 1 = S 0 ∪ S 1 . Note that the two points in S 1 do not have the same value in either coordinate.
We now consider the general case. Assume that for some 1 ≤ j ≤ J − 2 we have found sets S j , S j and S j−1 so that S j = S j−1 ∪ S j and i. | S j−1 | = |S j | = 2 j−1 and | S j | = 2 j . ii. Every point in S j satisfies condition (33). iii. No two points in S j have the same value in either coordinate.
We now construct the sets S j+1 and S j+1 . Consider the sets Lat j+1 (u) for u ∈ [0, 2 j+1 −1]∩Z. These 2 j+1 sets are disjoint and therefore at least 2 j of them do not contain any of the numbers that are the first coordinates of the points in S j . We let these sets correspond to u j+1 k ∈ [0, 2 j+1 − 1] ∩ Z for 1 ≤ k ≤ 2 j , and similarly, let v j+1 k ∈ [0, 2 j+1 − 1] ∩ Z, for 1 ≤ k ≤ 2 j , be so that no integer in Lat j+1 (v j+1 k ) coincide with the second coordinate of any point in S j . By Corollary 3.6, there exists a point in each of the sets Lat j+1 (u j+1 k , v j+1 k ) so that (33) holds. Put S j+1 to be the set containing all these points and let S j+1 = S j ∪ S j+1 . Note that S j+1 and S j+1 satisfy all of the conditions (i),(ii) and (iii) above, with j replaced by j + 1.
Now, for a fixed 0 ≤ j ≤ J − 1, each point (m, n) ∈ S j is of the form (m, n) = (u + σ (j) We observe that condition (33) implies the following for such a point (m, n) (where we apply the Cauchy-Schwarz inequality and (32) in the third step): The last step of the above computation follows by combining the observation that the summands are N-periodic, that is, |∆W (m, n)| = |∆W (m − N, n)| and |ΓW (m, n)| = |ΓW (m, n − N)|, with the fact that, by (32), the number of terms in each sum is bounded by 2 J , and therefore also by N.
Since no two points of S J−1 = ∪ J−1 j=0 S j have the same value in either coordinate, and |S j | = 2 j−1 , we obtain As J + 1 ≥ log N/ log 2, the desired lower inequality now follows.
Remark 4.4. Given the restriction N ≥ N 0 , it follows from Corollary 3.6 that the δ appearing in the last proof may be choosen to be δ = 2 √ A sin π(1/4 −1/N 0 ). Moreover, the inequality J + 1 ≥ log N/ log 2 yields Plugging this into the estimate (34), we find that For N 0 = 5, this yields the estimate c ≥ A/50, while as N 0 → ∞ we get that

4.3.
Proof for the upper bound in Theorem 4.2. We consider a function that first appeared in [5] (see also [23]). In these references, this function was used for similar purposes as here, namely, to provide examples of generators of orthonormal Gabor systems with close to optimal localisation. To define the function, we first let φ : R → [0, 1] denote a smooth function so that and γ : (0, 1] → [0, 1] a smooth function so that Using these functions, we define Finally, on [0, 1] 2 , we define the function G(x, y) = e 2πiH(x,y) , which we extend (continuously on R 2 \ Z 2 ) to a quasi-periodic function on all of R 2 (in a mild abuse of notation, we also denote the quasi-periodic extension by G(x, y)). Since the finite Zak transform is unitary, it follows that there exists a sequence b ∈ ℓ d 2 of unit norm so that In particular, since G is unimodular, it follows by Proposition 2.7 that the Gabor system G d (b) is an orthonormal basis for ℓ d 2 . The following proposition provides the required estimate from above on β 1,1 (N). .
To estimate (I), we make the following partition of the set [0, N − 1] 2 ∩ Z 2 : On A 0 , the values of G(m/N, n/N) are constant, so On A 2 , we use the fact that the function G is C ∞ on R 2 \Z 2 , and moreover, that on the set {(x, y) ∈ [0, 1] 2 : y ≥ x ∨ x ≥ 1/8} both G, and its derivatives, are continuous. Indeed, this means that we are justified in using the Mean Value theorem for (m, n) ∈ A 2 to make the estimate where µ m,n ∈ (m/N, (m + 1)/N) and C = C(G) > 0 is a constant not depending on N. It follows immediately that On A 1 , we do the computation where µ m,n ∈ (m/N, (m + 1)/N) as before. To estimate ( * ), we compute

This allows the bound
To estimate (II), we consider the corresponding sums over A 2 and over B = A 0 ∪ A 1 . We skip the estimates for A 2 , which are completely analogous to the corresponding estimates made above. Instead, we turn our focus to the estimate for B. As above, we begin by using the Mean Value Theorem to write This allows the estimate Remark 4.6. To determine a bound for the constant C of inequality (36), observe that we can actually choose φ to be piecewise linear, and therefore to satisfy φ ′ 2 ∞ ≤ 1. Moreover, observe that ( * ) + ( * * ) ≤ 8π 2 (C 0 + log N), where C 0 is some positive constant. Since the remaining sums over A 2 only contribute to C 0 , we conclude that asymptotically,

A quantitative Balian-Low type theorem in finite dimensions
In this section, we prove a finite dimensional version of the quantitative Balian-Low inequality. For the most part, we follow the main ideas appearing in our paper [24]. 5.1. Auxiliary results. The estimate in the following lemma is not optimal, but rather, chosen to simplify the presentation.  Proof. Fix b > 0 to be chosen later. We make the split .
To estimate (I), we compute We turn to estimate (II). Integrating by parts, we find that Rρ ′ (ξ)e 2πiξt dξ + 1 πt 2 2 sin(2πt) − sin(πt) . This yields the bound and the estimate We compute the value of b which minimizes the bound we obtained for (I)+(II) and find that this value is b = 30/7/π, which gives us the estimate (I) + (II) ≤ 9.67 < 10. (ii) φ, ψ ∈ ℓ d 2 such that n |∆φ(n)| ≤ 10R and n |∆ψ(n)| ≤ 10Q, Proof of Lemma 5.2. For the given A and for N 0 = 200, let δ 1 = 2 √ A sin(π/4 − π/200) be the constant from Corollary 3.6. Notice that the integers Q, R and N satisfy Therefore, there exist integers K and L that satisfy We choose K, L to be the smallest such integers. For s, t ∈ Z, let σ s and ω t be as defined in (23), that is, and ω t = tN L .

5.2.
A proof for Theorem 1.5. We are now ready to prove Theorem 1.5. In fact, we prove the more general version referred to in Remark 1.6 which we formulate as follows: Proof of Theorem 1.5. Let ρ : R → R be as in Lemma 5.1 and put Φ(t) = Rρ(Rt) and Ψ(t) = Qρ(Qt). Denote φ = S N P N Φ and ψ = S N P N Ψ. By Lemma 2.10, in combination with Lemma 5.1, it follows that n |∆φ(n)| ≤ 10R and n |∆ψ(n)| ≤ 10Q. As a consequence, the integers Q, R and N, as well as the functions φ, ψ and b, all satisfy the requirements of Lemma 5.2.
As Q, R < N/2, by Proposition 2.8, and Remark 2.9(i), . It therefore follows from Lemma 5.2, and the fact that both the finite Zak transform and the finite Fourier transform are unitary, that, for some constant C > 0, The result now follows by applying a suitable time-frequency translate to the sequence b.

Applications to the continuous setting
In this section we show that both the classical and quantitative Balian-Low theorems follow from their finite dimensional analogs. 6.1. Relating continuous and finite signals -revisited. We start by extending Proposition 2.8 to the space L 2 (R). We do this in four steps. In the first, we introduce some additional notations. To this end, fix N ∈ N, N ≥ 2, and let d = N 2 .
Step I: By (L 2 [0, 1/N] 2 ) d , we denote the space of all d-tuples {φ(j)} d−1 j=0 with function entries φ(j) ∈ L 2 ([0, 1/N] 2 ), equipped with the norm given by Note that the factor N appears in the norm in order to take the measure of Step II: We consider functions h(u, v; t) defined over [0, 1/N] 2 ×R, that are Nperiodic with respect to the variable t, and are such that, for every fixed t 0 , the restriction h(u, v; t 0 ) is well defined almost everywhere and belongs to L 2 ([0, 1/N] 2 ). Observe that the operator Above, we understand the notations F d S N h and Z d S N h to mean that F d and Z d operate on S N h with respect to the variable j with (u, v) being considered fixed.
Step III: Let f ∈ L 2 (R). For (u, v) ∈ [0, 1/N] 2 , we define the function and formally put Note that if f is in the Schwarz class S(R), then the function h(u, v; t) := P N f (u,v) (t) satisfies the conditions on h(u, v; t) described in Step II. The following lemma shows that this is true for all f ∈ L 2 (R).
Lemma 6.1. For f ∈ L 2 (R), let f (u,v) (t) be the function defined above. Then, for every fixed t 0 ∈ R, we have That is, the series defining P N f (u,v) (t 0 ) converges in the norm of L 2 ([0, 1/N] 2 ).
Proof. The lemma follows from the following computation, where, in the first step, we use the fact that { √ N e 2πiN ℓv } ℓ∈Z is an orthonormal basis over [0, 1/N]: It follows that, for f ∈ L 2 (R), the operator S N P N f (u,v) is well defined and that conditions (41), (42), (43) hold with h(u, v; t) = P N f (u,v) (t).
Step IV: For f in L 2 (R), we understand the notations F f (u,v) (t) and Zf (u,v) (t) to mean that the Fourier transform and the Zak transform are taken with respect to the variable t, with (u, v) being fixed. We now give our extension of Proposition 2.8. Proposition 6.2. Let f ∈ L 2 (R). Then, the following hold.
(i) For all N ∈ N and (m, n) ∈ Z 2 , we have where the equality holds in the sense of L 2 ([0, 1/N] 2 ). (ii) For all N ∈ N, we have where the equality holds in the sense of (L 2 [0, 1/N] 2 ) d .
Proof. First, Proposition 2.8 implies that (44) and (45) hold for f ∈ S(R) pointwise everywhere. Since S(R) is dense in L 2 (R), it is enough to show that the four operators implicitly defined by the left and right-hand sides of (44) and (45) are isometric (in fact, they are unitary).
To establish (45), we start by noting that T 1 : Since the finite Fourier transform F d is unitary, we conclude that F d •T 1 , implicitly defined on the left hand side of (45), is also unitary. Next, we note that u) . Since both the Fourier transform F and the operator T 1 are unitary, we conclude that the operator implicitly defined on the right hand side of (45) is also unitary, and therefore that (45) holds for all f ∈ L 2 (R).
To obtain (44) we first note that, since the discrete Zak transform is unitary, the above computation also implies that the operator defined by the left-hand side of (45), which acts from L 2 (R) to (L 2 [0, 1/N] 2 ) N ×N , is isometric. To complete the proof of Proposition 6.2, it therefore remains to show that the same is true for the operator T 2 : S(R) → (L 2 [0, 1/N] 2 ) N ×N , defined by ( To see this, we let f ∈ S(R), and make the following computation: In particular, we have the following. Corollary 6.3. Let g ∈ L 2 (R) be such that the Gabor system G(g) is a Riesz basis in L 2 (R) with lower and upper Riesz basis bounds A and B, respectively. Then, for almost every (u, v) ∈ [0, 1/N] 2 , the Gabor system G d (S N P N g (u,v) ) is a Riesz basis in ℓ d 2 with lower and upper Riesz basis bounds A and B satisfying A ≤ A and B ≤ B, respectively.
Proof. In light of Proposition 2.7, this result follows immediately from part (i) of Proposition 6.2.
6.2. The classical Balian-Low theorem. We start with the following lemma which relates the discrete and continuous derivatives of L 2 (R) functions.
where the integral on the right-hand side is understood to be infinite if f is not absolutely continuous, or if its derivative is not in L 2 (R).
Proof. Put a u (j, l) = f (u + j N + ℓN) and fix j ∈ [0, N 2 − 1] ∩ Z. Since f ∈ L 2 (R), we have, with respect to the variable ℓ, that a u (j, ℓ) ∈ ℓ 2 (Z) for almost every u. We compute where we applied the inequality |a + b| 2 ≤ 2|a| 2 + 2|b| 2 in the second step and the inequality |e ix − 1| ≤ |x| in the third step. By the Cauchy-Schwartz inequality, we get Combining these two estimates we find that We are now ready to show that the classical Balian-Low theorem (Theorem A) follows from our finite Balian-Low theorem (Theorem 1.1).
Proof of the Classical Balian-Low Theorem. Let g ∈ L 2 (R) be such that the Gabor system G(g) is a Riesz basis with lower and upper Riesz basis bounds A and B, respectively. For all integers N ≥ 5, d = N 2 , and u, v ∈ [0, 1/N] 2 , we consider the finite dimensional signal S N P N g (u,v) . By Corollary 6.3, for almost every u, v ∈ [0, 1/N] 2 , this is a basis in ℓ d 2 with Riesz basis bounds A, B satisfying A ≤ A and B ≤ B. By Theorem 1.1, (see also Remark 1.2) we have Integrating both parts with respect to (u, v) over the set [0, 1/N] 2 , and applying Proposition 6.2 (ii), we get By Lemma 6.4, we obtain Finally, letting N tend to infinity, the result follows.
6.3. A quantiative Balian-Low theorem. Discrete and continuous tail estimates are related by the following lemma.
Lemma 6.5. Let f ∈ L 2 (R) and Q, N ∈ N be such that Q ≤ N. Denote F (u,v) = S N P N f (u,v) . Then, Proof. We have, We are now ready to show that the Quantitative Balian-Low theorem (Theorem B) follows from Theorem 1.5 (or rather, the more general Theorem 5.3).
Proof of the Quantitative Balian-Low Theorem. Let g ∈ L 2 (R) be such that G(g) is a Riesz basis with lower and upper Riesz basis bounds A and B, respectively, and let Q, R be positive integers. Let N ≥ 200 max{Q, R}· B/A and set d = N 2 . For fixed u, v ∈ R, consider the finite dimensional signal S N P N g (u,v) . By Corollary 6.3, for almost every u, v ∈ [0, 1/N] 2 , this is a basis in ℓ d 2 with Riesz basis bounds A, B satisfying A ≤ A and B ≤ B. By Theorem 5.3 we have Integrating both terms with respect to (u, v) over the set [0, 1/N] 2 , and applying Proposition 6.2 (ii), we get By Lemma 6.5 we obtain, The result now follows by applying an appropriate translation and modulation to g (note that a translations and modulations of g preserve the Riesz basis properties of G(g)).

Balian-Low theorems for Gabor systems over rectangles
In this section, we turn to the proofs of theorems 1.9 and 1.10, which consider finite Gabor systems over rectangular lattices. Note that, as the proofs are very similar to those of theorems 1.1 and 1.5, respectively, we only provide the necessary preliminaries and an outline of the main ideas. For easy reference, we give the subsections below titles that match those of the discussion in the square lattice case.
With the normalization it is readily checked that the Fourier transform is a unitary operator from ℓ The finite Zak transform for a ∈ ℓ (M,N ) 2 is given by Note that with this definition, Z (M,N ) (a) is well-defined as a function on Z 2 d (that is, it is d-periodic separately in each variable).
The basic properties of Z (M,N ) are stated in the following lemma (compare with Lemma 2.2).  where the subscript of * 1 indicates that the convolution is taken with respect to the first variable of the finite Zak transform.
An extension of part (ii) of Proposition 2.7 to the rectangular case reads as follows.  The following result extends proposition 2.8.
for integers K ∈ [2, M] and L ∈ [2, N]. Note that, similar to the square case, we have σ K = M and ω L = N. With this we get the following extension of Corollary 3.6.
It follows that Theorem 1.9 is a consequence of the following result. (58) By repeating the same type of argument for the case M < N, the estimate from below in (57) follows.