# Balian–Low type theorems in finite dimensions

- 150 Downloads

## Abstract

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.

## Mathematics Subject Classification

42C15 42A38 39A12## 1 Introduction

### 1.1 A Balian–Low type theorem in finite dimensions

*g*with respect to the lattice \(\mathbb {Z}^2\) is given by

*G*(

*g*) is an orthonormal basis, or a Riesz basis, in \(L^2(\mathbb {R})\), 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

*G*(

*g*) is an orthonormal basis or a Riesz basis in \(L^2(\mathbb {R})\), then

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.

*g*be a continuous function in \(L^2(\mathbb {R})\) and put \(b(j) = g({j}/{N})\), \(j\in \mathbb {Z}\cap [-N^2/2,N^2/2)\). That is, the sequence \(b \in \ell _2^d\) 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 \(\ell ^2\) norm can be interpreted as a Riemann sum approximating the \(L^2(\mathbb {R})\) norm of

*g*. Note that in Sect. 2, we define the finite Fourier transform \({\mathcal {F}}_d\) so that it is unitary on \(\ell _2^d\).

*d*. For \(b\in \ell _2^d\), the Gabor system generated by

*b*with respect to \((N\mathbb {Z}_d)^2\), is given by

*G*(

*g*) restricted to the interval \([-N/2,N/2)\).

*g*at the points

*j*/

*N*are approximated by \(N\Delta b\). 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

*N*tends to infinity.

### Remark 1.2

Theorem 1.1 also holds in the case that the infimum in \(\alpha (N)\) is taken over all \(b\in \ell _2^d\) for which the system \(G_d(b)\) is a basis in \(\ell _2^d\) 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 Sect. 2). The dependence on the Riesz basis bounds is necessary, in the sense that it can not be replaced by a dependence on the \(\ell _2^d\) norm of *b* (see Remark 4.3).

### Remark 1.3

The classical Balian–Low theorem (Theorem A) follows as a corollary of Theorem 1.1, as we show in Sect. 6.

### Remark 1.4

### 1.2 A finite dimensional quantitative Balian–Low type theorem

### Theorem B

*G*(

*g*) is an orthonormal basis or a Riesz basis then, for every \(Q,R>1\), we have

*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 non-symmetric cases and the amalgam space cases referred to above. Here, we prove the following finite dimensional version of this theorem.

### Theorem 1.5

### Remark 1.6

The comments made in Remarks 1.2 and 1.3 hold also for Theorem 1.5 (see Theorem 5.3 and Sect. 6.3, respectively).

### Remark 1.7

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 \(N \ge {350}\). 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 \(N \ge {350}\) and \(Q,R \le N/{30}\) 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 \(\mathbb {Z}^2\) by Gabor systems over the rectangular lattices \(\lambda \mathbb {Z}\times \frac{1}{\lambda }\mathbb {Z}\), where \(\lambda >0\). 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 Theorems 1.1 and 1.5 formulated below.

*g*is a continuous function in \(L^2(\mathbb {R})\) and \(b(j) = g({j}/{ M} )\), \(j\in \mathbb {Z}\cap [-MN/2,MN/2)\), then the above \(\ell ^2\)-norm can be interpreted as a Riemann sum for the \(L^2(\mathbb {R})\) norm of

*g*over the interval \([-N/2, N/2]\). Note that in Sect. 7, we define the finite Fourier transform \({\mathcal {F}}_{(M,N)}\) so that it is a unitary operator from \(\ell _2^{(M,N)}\) to \(\ell _2^{(N,M)}\).

*b*with respect to \(M\mathbb {Z}_{d} \times N\mathbb {Z}_{d}\) is given by

*M*–discretization of the Gabor system

*G*(

*g*) restricted to \([-N/2,N/2]\).

We are now ready to formulate the extensions of Theorems 1.1 and 1.5 to Gabor systems over rectangles.

### Theorem 1.8

### Theorem 1.9

We point out that Remarks 1.2 and 1.6 also hold for Theorems 1.8 and 1.9, respectively. That is, these theorems can be extended to generators of general bases with the constants depending only on the Riesz basis bounds. Remark 1.7 also holds in this case.

### 1.4 The structure of the paper

In Sect. 2 we discuss some preliminaries, in particular the finite and continuous Zak transform. In Sect. 3 we present two improved versions of a lemma that we first proved in [24]. These results quantify the discontinuity of the argument of a quasi-periodic function. In Sect. 4, we apply these lemmas to prove Theorem 1.1, while in Sect. 5, we use them to prove Theorem 1.5. In Sect. 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 Sect. 7 we discuss Theorems 1.8 and 1.9. We give only a sketch of a proof for Theorem 1.8, as the proofs in the rectangular case are very similar to the proofs in the square lattice case.

## 2 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 \(\ell ^d_2\), and by *b* a function in \(\ell ^d_2\) which is a generator of a Gabor system.

Also, recall that for a sequence \(a\in \ell _2^d\), we denote the discrete derivative by \(\Delta a(j) = a(j+1) - a(j)\). From time to time, we encounter sequences depending on more than one variable, say \(a\big (k + \psi (s)\big )\) where \(\psi \) is some function depending on the integer *s*. In this case, we write \(\Delta _{(s)}\) if we want to indicate that the difference is to be taken with respect to *s*. That is, \(\Delta _{(s)} a\big (k+ \psi (s)\big ) = a\big (k + \psi (s+1)\big ) - a\big (k + \psi (s)\big )\).

*N*, and computing the corresponding Rieman sum.

### 2.2 The continuous and finite Zak transforms

On \(L^2(\mathbb {R})\), the Zak transform is defined as follows.

### Definition 2.1

*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].

### Lemma 2.2

- i.The Zak transform is quasi-periodic on \(\mathbb {R}^2\) in the sense thatIn particular, this means that the function$$\begin{aligned} Zf(x+1,y) = \mathrm {e}^{2\pi \mathrm {i}y} Zf(x,y) \quad \text {and} \quad Zf(x,y+1) = Zf(x,y). \end{aligned}$$(11)
*Zf*is determined by its values on \([0,1]^2\). - ii.
The Zak transform is a unitary operator from \(L^2(\mathbb {R})\) onto \(L^2([0,1]^2)\).

- iii.The Zak transform and the Fourier transform satisfy the relation$$\begin{aligned} Z ({\mathcal {F}}f)(x,y) = \mathrm {e}^{2\pi \mathrm {i}xy} Zf(-y,x). \end{aligned}$$
- iv.For \(\phi \in {\mathcal {S}}(\mathbb {R})\), the Zak transform satisfies the convolution relationwhere the subscript of \(*_1\) indicates that the convolution is taken with respect to the first variable of the Zak transform.$$\begin{aligned} Z (f *\phi ) = Z (f) *_1 \phi , \end{aligned}$$

Next, we discuss a Zak transform for \(\ell _2^d\) which appears in, e.g., [2].

### Definition 2.3

*finite Zak transform*of \(a\in \ell _2^d\), with respect to \((N\mathbb {Z}_d)^2\), is given by

Note that with this definition, \(Z_d(a)\) is well-defined as a function on \(\mathbb {Z}_d^2\) (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.

### Lemma 2.4

- (i)The function \(Z_d(a)\) is
*N*-quasi-periodic on \(\mathbb {Z}_d^2\) in the sense thatIn particular, \(Z_d(a)\) is determined by its values on the set \(\mathbb {Z}^2 \cap [0,N-1]^2\).$$\begin{aligned} \begin{aligned} Z_d(a)(m + N,n)&= \mathrm {e}^{2\pi \mathrm {i}\frac{n}{N}}Z_d(a)(m,n), \\ Z_d(a)(m,n+N)&= Z_d (a)\,(m,n). \end{aligned} \end{aligned}$$(12) - (ii)
The transform \(Z_d\) is a unitary operator from \(\ell _2^d\) onto \(\ell _2 ([0,N-1]^2)\).

- (iii)The finite Zak transform and the finite Fourier transform satisfy the relation$$\begin{aligned} Z_d ({\mathcal {F}}_d a)(m,n) = \, \mathrm {e}^{2\pi \mathrm {i}\frac{mn}{d}} Z_d(a) (-n,m). \end{aligned}$$(13)
- (iv)The finite Zak transform satisfies the convolution relationwhere the subscript of \(*_1\) indicates that the convolution is taken with respect to the first variable of the finite Zak transform.$$\begin{aligned} Z_d (a *\phi ) = Z_d(a) *_1 \phi ,\qquad a, \phi \in \ell _2^d. \end{aligned}$$

### Remark 2.5

*N*-quasi periodicity. Namely, we will be interested in functions \(W : \mathbb {Z}^2_d \rightarrow \mathbb {C}\) satisfying

*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

### Proof

For a function \(a\in \ell _2^d\), write \(\Delta _ka(n)=a(n+k)-a(n)\). Property (iv) of Lemma 2.4 implies that \(\Delta _k Z_d(a *\phi )=Z_d(a)*_1(\Delta _k \phi )\). The result now follows by applying the triangle inequality to \(\sum |\Delta _k \phi |\). \(\square \)

### 2.3 Gabor Riesz bases and the Zak transform

*H*is called a

*Riesz basis*if it is the image of an orthonormal basis under a bounded and invertible linear transformation \(T:H\mapsto H\). Equivalently, the system \(\{f_n\}\) is a Riesz basis if and only if it is complete in

*H*and satisfies the inequality

*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, Corollary 8.3.2(b)], while part (ii) can be found in [2, Theorem 6].

### Proposition 2.7

- (i)
Let \(g\in L^2(\mathbb {R})\). Then,

*G*(*g*) is a Riesz basis in \(L^2(\mathbb {R})\) with Riesz basis bounds*A*and*B*if and only if \(A\le |Zg(x,y)|^2\le B\) for almost every \((x,y)\in [0,1]^2\). - (ii)
Let \(N\in \mathbb {N}\), \(d=N^2\) and \(b\in \ell _2^d\). Then, \(G_d(b)\) is a basis in \(\ell _2^d\) with Riesz basis bounds

*A*and*B*if and only if \(A\le |Z_d(b)(m,n)|^2\le B\) for all \((m,n) \in [0,N-1]^2 \cap \mathbb {Z}^2\).

### 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.

*f*in the Schwartz class \({\mathcal {S}}(\mathbb {R})\) and a continuous

*N*-periodic function

*h*, we define their

*N*-periodisation and

*N*-samples, respectively, by

### Proposition 2.8

*f*in the Schwartz class \({\mathcal {S}}(\mathbb {R})\), the following hold.

- (i)For every \(N\in \mathbb {N}\) and \((m,n)\in \mathbb {Z}^2\), we have$$\begin{aligned} Z_{d} (S_N {P_N f})(m,n) = Zf(m/N, n/N). \end{aligned}$$(17)
- (ii)For every \(N\in \mathbb {N}\), we have$$\begin{aligned} {\mathcal {F}}_{d}S_N P_N f = S_N P_N {\mathcal {F}} f. \end{aligned}$$(18)

### Proof

(i): Since \(f \in {\mathcal {S}}(\mathbb {R})\), we can change the order of summation in the expression for \(Z_d(S_N P_N f)(m,n)\). This immediately yields the formula.

(ii): Observe that part (i) holds for both *f* and \({\mathcal {F}}f\). With this, in combination with parts (iii) of Lemmas 2.2 and 2.4, the proof of part (ii) follows. \(\square \)

### Remark 2.9

- (i)
Although Proposition 2.8 is formulated for functions in the Schwartz class \({\mathcal {S}}(\mathbb {R})\), it is readily checked that it holds for all functions \(f \in L^2(\mathbb {R})\) which satisfy both \(\sup _{t \in \mathbb {R}} |t^2 f(t)| < \infty \) and \(\sup _{\xi \in \mathbb {R}} |\xi ^2 {{\hat{f}}}(\xi )| < \infty \).

- (ii)

The following lemma relates the discrete and continuous derivatives.

### Lemma 2.10

### Proof

The inequality follows by applying the Fundamental Theorem of Calculus to the differences \(\Delta a_j\) and then applying the triangle inequality. \(\square \)

## 3 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 \(\mathbb {Z}_d^2\)

### Lemma 3.1

*H*be a function on \(([0,K] \times [0,L]) \cap \mathbb {Z}^2\) that satisfies

### 3.2 ‘Jumps’ of quasi-periodic functions on subsets of \(\mathbb {Z}_d^2\)

*a*] denotes the integer part of

*a*. Note that \(\sigma _K=\omega _L=N\). We can now state the following lemma.

### Lemma 3.4

*W*be a function defined over \(\mathbb {Z}_d^2\) that is

*N*-quasi-periodic up to a constant (see (14)). Denote by

*H*any branch of the argument of

*W*, so that \(W= |W| \mathrm {e}^{2\pi \mathrm {i}H}\). Then, there exist \((s,t) \in ([0,K-1]\times [0,L-1]) \cap \mathbb {Z}^2\) so that either

### Proof

*W*with the constant \(\eta =e^{2\pi i \gamma }\). We start by modifying the argument \(H(\sigma _{s}, \omega _{t})\) of

*W*to obtain a function that satisfies the conditions of Lemma 3.1. To this end, for \((s,t)\in \mathbb {Z}^2\), set \(h_{s,t} = H(\sigma _{s}, \omega _{t})\), and define a function \(\Phi (s,t)\) on \(([0,K]\times [0,L]) \cap \mathbb {Z}^2\) as follows:

*H*is the argument of a function that is

*N*-periodic in the second variable, we have

*s*,

*t*) with \(s \le K-2\), then this is immediate from the definition of \(\Phi \). If the jump is in the horizontal direction, at a point (

*s*,

*t*) with \(s = K-1\), then we note that, since \(\sigma _K = N\), the

*N*-quasi-periodicity up to a constant of

*W*implies that

We obtain the following corollary of Lemma 3.4.

### Corollary 3.5

*W*is an

*N*-quasi periodic function satisfying \(A \le |W|^2\) over the lattice \(\mathbb {Z}^2_d\), then, for every \((u, v) \in \mathbb {Z}_d^2\), there exists at least one point \((s,t)\in ([0,K-1] \times [0,L-1])\cap \mathbb {Z}^2\) such that

### Proof

First, we point out that we define the argument \(\arg (z)\) of a complex number *z* so that \(z=|z|e^{2\pi i\text {arg}(z)}\). 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 \(N\times N\) square the argument of *W* jumps by more than \({{1}/{8}} - 1/N\) in at least one of the inequalities (24) or (25). As the modulus of *W* is bounded from below by \(\sqrt{A}\), the conclusion now follows from basic trigonometry. \(\square \)

## 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

*G*(

*b*) is an orthonormal basis.

### Proposition 4.1

### Proof

We will only prove the right-hand side inequality in (26) since the left-hand side inequality is proved in the same way.

*N*-periodic in the second variable, we find that

*A*and at most

*B*.

### Theorem 4.2

### Remark 4.3

*Zh*has exactly one zero on the unit square located at \((1/2+\tau ,1/2)\). Since the first coordinate of this point is irrational, it follows by Proposition 2.8 that the function \(Z_d(b_{N})\) with \(b_{N} = S_N P_N h\), where \(d=N^2\), does not have a zero on \(\mathbb {Z}_d^2\). Consequently, Proposition 2.7 implies that the finite Gabor system \(G_d(b_N)\) is a basis for \(\ell ^d_2\), though the (lower) Riesz basis bounds of these bases decay as

*N*increases. Straight-forward computations, using only the regularity and decay of the Gaussian, show that there exist constants \(C,D, E>0\), such that the following hold:

- (i)
\(C \le \Vert b_N\Vert _d\le D\) for every \(N\in \mathbb {N},\) (ii) \(\alpha (b_N,N)\le E\) as \(N \rightarrow \infty \).

*c*in Theorem 1.1 depends on the lower Riesz basis bounds in the definition of \(\alpha _{A,B}(N)\). Similarly, it may be shown that

*C*depends on the upper Riesz basis bound.

### 4.2 Proof for the lower bound in Theorem 4.2

### Proof

*W*‘jumps’, i.e., where

*W*that are, in some sense, separated. We do this in an inductive process.

In the first step, let \(j=0\). By Corollary 3.5, there exists a point \((m_0,n_0)\) in \(\mathrm {Lat}_0(0,0)\) so that (31) holds for this point. Let \({\widetilde{S}}_0=S_0 = \{(m_0,n_0)\}\). Next, let \(j=1\). For \(u\in \{0,1\}\), the sets \(\mathrm {Lat}_1(u)\) are disjoint, and so at least one of them does not contain the number \(m_0\). Let \(u^{1}_1 \in \{0,1\}\) be such that the set \(\mathrm {Lat}_1(u_1^1)\) has this property, and, similarly, let \(v^{1}_1\in \{0,1\}\) be such that \(\mathrm {Lat}_1(v^{1}_1)\) does not contain the number \(n_0\). By Corollary 3.5, there exists a point \((m_1,n_1)\) in \(\mathrm {Lat}_{1}(u^{1}_1,v^{1}_1)\) so that (31) holds for this point. Let \(S_1 = \{(m_1,n_1)\}\), and put \(\widetilde{S_1}=S_0\cup S_1\). Note that the two points in \({\widetilde{S}}_1\) do not have the same value in either coordinate.

- i.
\(|{\widetilde{S}}_{j-1}|= |S_j|=2^{j-1}\) and \(|{\widetilde{S}}_j|=2^j\).

- ii.
Every point in \({S}_j\) satisfies condition (31).

- iii.
No two points in \({\widetilde{S}}_j\) have the same value in either coordinate.

*j*replaced by \(j+1\).

*m*,

*n*) (where we apply the Cauchy–Schwarz inequality and (30) in the third step):

*N*-periodic, that is, \(|\Delta W(m,n)| = |\Delta W(m-N,n)|\) and \(|\Gamma W(m,n)| = |\Gamma W(m,n-N)|\), with the fact that, by (30), the number of terms in each sum is bounded by \(2^J\), and therefore also by

*N*.

### Remark 4.4

### 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.

*G*(

*x*,

*y*)). Since the finite Zak transform is unitary, it follows that there exists a sequence \(b \in \ell ^d_2\) of unit norm so that

*G*is unimodular, it follows by Proposition 2.7 that the Gabor system \(G_d(b)\) is an orthonormal basis for \(\ell _2^d\).

The following proposition provides the required estimate from above on \(\beta _{1,1}(N)\).

### Proposition 4.5

*b*be the sequence defined above. Then, there exists a constant \(C>0\) so that for all \(N \ge 2\) and \(d = N^2\), we have

### Proof

*C*positive constants which may change from line to line. In light of (33), we need to estimate the expression

*I*), we make the following partition of the set \( [0,N-1]^2 \cap \mathbb {Z}^2\):

*G*(

*m*/

*N*,

*n*/

*N*) are constant, so

*G*is \(C^\infty \) on \(\mathbb {R}^2 \backslash \mathbb {Z}^2\), and moreover, that on the set \(\{(x,y)\in [0,1]^2:y\ge x\vee x\ge 1/8\}\) both

*G*, and its derivatives, are continuous. Indeed, this means that we are justified in using the Mean Value theorem for \((m,n) \in A_2\) to make the estimate

*N*. It follows immediately that

*B*. As above, we begin by using the Mean Value Theorem to write

### Remark 4.6

*C*of inequality (34), observe that we can actually choose \(\phi \) to be piecewise linear, and therefore to satisfy \(\Vert \phi '\Vert _\infty ^2 \le 1\). Moreover, observe that

## 5 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. We leave the proof, which is straight-forward, to the reader.

### Lemma 5.1

### Lemma 5.2

- (i)
\(Q, R \in \mathbb {Z}\) such that \(1\le Q,R \le (N/{30}) \cdot \sqrt{A/B}\),

- (ii)
\(\phi ,\psi \in \ell _2^d\) such that \(\sum _n|\Delta \phi (n)|\le 10 R\) and \(\sum _n|\Delta \psi (n)|\le 10 Q\),

- (iii)
\(b\in \ell _2^d\) such that \(A \le |Z_d (b)|^2 \le B\).

### Proof of Lemma 5.2

*A*and for \(N_0={350}\), let \(\delta _1 = 2\sqrt{A}\sin (\pi /8-\pi /{350})\) be the constant from Corollary 3.5. Notice that the integers

*Q*,

*R*and

*N*satisfy

*K*and

*L*that satisfy

*K*,

*L*to be the smallest such integers.

*K*and

*L*imply that, for some constant \(C_1=C_1(A,B)\),

*u*,

*v*), either the set \(\mathrm {Lat}(u,v)\) or the set \(\mathrm {Lat^*}(u,v)\) contains a point from \(([0,N-1]\cap \mathbb {Z})^2\) which satisfies condition (35) or (36), respectively. Putting \(C=C_1/2\) our proof will then be complete.

So, fix \((u,v)\in ([0,\Sigma -1]\cap \mathbb {Z})\times ([0,\Omega -1]\cap \mathbb {Z})\). Due to Corollary 3.5, there exists at least one point \((s,t)\in ([0,K-1]\cap \mathbb {Z})\times ([0,L-1]\cap \mathbb {Z})\) such that (24) or (25) hold with \(W= Z_d(b)\).

*N*-quasi-periodicity of the Zak transform, the point \((u,v+\omega _t)\) satisfies the same inequality, and is in \(([0,N-1]\cap \mathbb {Z})^2\).

*u*,

*v*) and (

*s*,

*t*). Then (13), the

*N*-quasi-periodicity of \(Z_d(b)\), and the estimate \(|\Delta _{(t)}\omega _t|\le {2N/L}\), imply that

*N*-quasi-periodicity of the Zak transform, the point \((N-v,u+\sigma _s)\) satisfies the same inequality, and is in \(([0,N-1]\cap \mathbb {Z})^2\). \(\square \)

### 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:

### Theorem 5.3

*A*and

*B*. Then, for all positive integers \(1 \le Q, R \le ( N/{30})\cdot \sqrt{A/B}\), we have

### Proof of Theorem 1.5

Let \(\rho : \mathbb {R}\rightarrow {\mathbb {R}}\) be as in Lemma 5.1 and put \(\Phi (t) = R \rho (Rt)\) and \(\Psi (t) = Q \rho ( Qt)\). Denote \(\phi = S_N P_N \Phi \) and \(\psi = S_N P_N \Psi \). By Lemma 2.10, in combination with Lemma 5.1, it follows that \(\sum _n|\Delta \phi (n)|\le {10} R\) and \(\sum _n|\Delta \psi (n)|\le {10} Q\). As a consequence, the integers *Q*, *R* and *N*, as well as the functions \(\phi ,\psi \) and *b*, all satisfy the requirements of Lemma 5.2.

*b*. \(\square \)

## 6 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 \(L^2(\mathbb {R})\). We do this in four steps. In the first, we introduce some additional notations. To this end, fix \(N\in \mathbb {N}\), \(N\ge 2\), and let \(d=N^2\).

**Step I:**By \((L^2[0,1/N]^2)^d\), we denote the space of all d-tuples \(\{ \phi (j)\}_{j=0}^{d-1}\) with function entries \(\phi (j) \in L^2([0, {1}/{N}]^2)\), equipped with the norm given by

*N*appears in the norm in order to take the measure of [0, 1 /

*N*] into account. Similarly, by \((L^2[0,1/N]^2)^{N\times N}\), we denote the space of all \(N\times N\) matrices \(\{ \phi (j,k)\}_{j,k=0}^{N-1}\) with function entries \(\phi (j,k) \in L^2([0, {1}/{N}]^2)\), equipped with norm given by

**Step II:**We consider functions

*h*(

*u*,

*v*;

*t*) defined over \([0, {1}/{N}]^2\times \mathbb {R}\), that are

*N*-periodic 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

*j*with (

*u*,

*v*) being considered fixed.

**Step III:**Let \(f\in L^2(\mathbb {R})\). For \((u,v)\in [0, {1}/{N}]^2\), we define the function

*f*is in the Schwarz class \({\mathcal {S}}(\mathbb {R})\), 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 \(f\in L^2(\mathbb {R})\).

### Lemma 6.1

### Proof

Use the fact that \(\{\sqrt{N}\mathrm {e}^{2\pi \mathrm {i}N\ell v}\}_{\ell \in \mathbb {Z}}\) is an orthonormal basis over [0, 1 / *N*]. \(\square \)

It follows that, for \(f\in L^2(\mathbb {R})\), the operator \(S_NP_Nf_{(u,v)}\) is well defined and that conditions (40), (41), (42) hold with \(h (u,v;t)=P_Nf_{(u,v)}(t)\).

**Step IV:** For *f* in \(L^2(\mathbb {R})\), we understand the notations \({\mathcal {F}} f_{(u,v)}(t)\) and \(Z f_{(u,v)}(t)\) to mean that the Fourier transform and the Zak transform are taken with respect to the variable *t*, with (*u*, *v*) being fixed. Since \({\mathcal {S}}(\mathbb {R})\) is dense in \(L^2(\mathbb {R})\), we obtain the following extension of Proposition 2.8.

### Proposition 6.2

- (i)For all \(N\in \mathbb {N}\) and \((m,n)\in \mathbb {Z}^2\), we havewhere the equality holds in the sense of \({L^2([0,{1}/{N}]^2)}\).$$\begin{aligned} Z_{d} (S_N {P_N f_{(u,v)}})(m,n) = Zf_{(u,v)}(m/N, n/N), \end{aligned}$$(43)
- (ii)For all \(N\in \mathbb {N}\), we havewhere the equality holds in the sense of \((L^2[0,{1}/{N}]^2)^d\).$$\begin{aligned} {\mathcal {F}}_{d}S_N P_N {f_{(u,v)}} = S_N P_N {\mathcal {F}} f_{(u,v)}, \end{aligned}$$(44)

### Proof

First, Proposition 2.8 implies that (43) and (44) hold for \(f \in {\mathcal {S}}(\mathbb {R})\) pointwise everywhere. Since \({\mathcal {S}}(\mathbb {R})\) is dense in \(L^2(\mathbb {R})\), it is enough to show that the four operators implicitly defined by the left and right-hand sides of (43) and (44) are isometric (in fact, they are unitary). This easily follows by using the fact that \(\{\sqrt{N}\mathrm {e}^{2\pi \mathrm {i}N\ell v}\}_{\ell \in \mathbb {Z}}\) is an orthonormal basis over [0, 1 / *N*]. \(\square \)

In light of Proposition 2.7, we get the following from part (i) of Proposition

### Corollary 6.3

Let \(g\in L^2(\mathbb {R})\) be such that the Gabor system *G*(*g*) is a Riesz basis in \(L^2(\mathbb {R})\) with lower and upper Riesz basis bounds *A* and *B*, respectively. Then, for almost every \((u,v)\in [0,{1}/{N}]^2\), the Gabor system \(G_d(S_NP_N g_{(u,v)})\) is a Riesz basis in \(\ell _2^d\) with lower and upper Riesz basis bounds \({{\widetilde{A}}}\) and \({{\widetilde{B}}}\) satisfying \(A \le {{\widetilde{A}}}\) and \(\widetilde{B} \le B\), respectively.

### 6.2 The classical Balian–Low theorem

We start with the following lemma which relates the discrete and continuous derivatives of \(L^2(\mathbb {R})\) functions.

### Lemma 6.4

*f*is not absolutely continuous, or if its derivative is not in \(L^2(\mathbb {R})\).

### Proof

*u*. We compute

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

*G*(

*g*) is a Riesz basis with lower and upper Riesz basis bounds

*A*and

*B*, respectively. For all integers \(N\ge 2\), \(d = N^2\), and \(u,v\in [0,1/N]^2\), we consider the finite dimensional signal \(S_NP_N g_{(u,v)}\). By Corollary 6.3, for almost every \(u,v\in [0,1/N]^2\), this is a basis in \(\ell _2^{d}\) with Riesz basis bounds \({{\widetilde{A}}}\), \({{\widetilde{B}}}\) satisfying \(A \le {{\widetilde{A}}}\) and \({{\widetilde{B}}} \le B\). By Theorem 1.1, (see also Remark 1.2) we have

*u*,

*v*) over the set \([0, {1}/{N}]^2\), and applying Proposition 6.2 (ii), we get

*N*tend to infinity, the result follows. \(\square \)

### 6.3 A quantiative Balian–Low theorem

Discrete and continuous tail estimates are related by the following lemma.

### Lemma 6.5

### Proof

This follows by using the fact that \(\{\sqrt{N}\mathrm {e}^{2\pi \mathrm {i}N\ell v}\}_{\ell \in \mathbb {Z}}\) is an orthonormal basis over [0, 1 / *N*]. \(\square \)

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

*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 \(N\ge {200} \max \{Q,R\}\cdot \sqrt{B/A}\) and set \(d = N^2\). For fixed \(u,v\in \mathbb {R}\), consider the finite dimensional signal \(S_NP_Ng_{(u,v)}\). By Corollary 6.3, for almost every \(u,v\in [0,1/N]^2\), this is a basis in \(\ell _2^{d}\) with Riesz basis bounds \(\widetilde{A}\), \({{\widetilde{B}}}\) satisfying \(A \le {{\widetilde{A}}}\) and \(\widetilde{B} \le B\). By Theorem 5.3 we have

*u*,

*v*) over the set \([0,{1}/{N}]^2\), and applying Proposition 6.2 (ii), we get

*g*(note that a translations and modulations of

*g*preserve the Riesz basis properties of

*G*(

*g*)). \(\square \)

## 7 Balian–Low theorems for Gabor systems over rectangles

In this section, we turn to theorems 1.8 and 1.9. As the proofs are very similar to those of theorems 1.1 and 1.5, respectively, we only provide an outline of the main ideas, and leave it to the reader to fill in the details.

### 7.1 The finite Zak transforms over rectangles.

*d*-periodic separately in each variable).

The basic properties of \(Z_{(M,N)}\) are stated in the following two results (compare with Lemma 2.2 and part (ii) of Proposition 2.7).

### Lemma 7.1

- (i)The function \(Z_{(M,N)}(a)\) is (
*M*,*N*)-quasi-periodic on \(\mathbb {Z}_d^2\) in the sense thatIn particular, the finite Zak transform is determined by its values on the set \(\mathbb {Z}^2 \cap ([0,M-1]\times [0,N-1])\).$$\begin{aligned} \begin{aligned} Z_{(M,N)}(a)(m + M,n)&= \mathrm {e}^{2\pi \mathrm {i}\frac{n}{N}}Z_{(M,N)}(a)(m,n), \\ Z_{(M,N)}(a)(m,n+N)&= Z_{(M,N)} (a)(m,n). \end{aligned} \end{aligned}$$(45) - (ii)
The transform \(Z_{(M,N)}\) is a unitary operator from \(\ell _2^{(M,N)}\) onto \(\ell _2([0,M-1]\times [0,N-1])\).

- (iii)The finite Zak transform and the finite Fourier transform satisfy the relation$$\begin{aligned} Z_{(N,M)} ({\mathcal {F}}_{(M,N)}a)(n,m) = \, \mathrm {e}^{2\pi \mathrm {i}\frac{mn}{d}} Z_{(M,N)}(a) (-m,n). \end{aligned}$$(46)

### Proposition 7.2

Let \(M,N\in \mathbb {N}\) and \(b\in \ell _2^{(M,N)}\). Then the Gabor system \(G_{(M,N)}(b)\), defined in (9), is a basis in \(\ell _2^{(M,N)}\) with Riesz basis bounds *A* and *B*, if and only if \(A\le |Z_{(M,N)}(b)(m,n)|^2\le B\) for all \((m,n) \in \mathbb {Z}^2\cap ([0,M-1]\times [0,N-1])\).

### 7.2 Regularity of the finite Zak transforms over rectangles

### Corollary 7.3

*W*be an (

*M*,

*N*)-quasi periodic function over the lattice \(\mathbb {Z}_d^2\) satisfying \(A \le |W(m,n)|^2\), for all \((m,n)\in \mathbb {Z}^2_d\). Then, for every \((u, v) \in \mathbb {Z}_d^2\) there exists at least one point \((s,t)\in ([0,K-1] \times [0,L-1]) \cap \mathbb {Z}^2\) such that

### 7.3 A Balian–Low type theorem in finite dimensions over rectangles

*A*and

*B*, respectively. The analog of Proposition 4.1 for the rectangular case gives the inequalities

### Theorem 7.4

### Proof

*c*,

*C*denote different constants which may change from line to line. Let \(b\in \ell ^{(M,N)}_2\) be so that \(G_{(M,N)}(b)\) is a basis with Riesz basis bounds

*A*and

*B*, and put \(W = Z_{(M,N)}(b)\). To obtain the lower inequality, we first consider the case \(M>N\). Put \(\sigma _s = [s M/N]\), and consider square product sets

*W*is N–quasi–periodic, it follows by Theorem 4.2 that

*M*-periodic. Plugging this into (53) we get,

*M*/

*N*] sets \(\mathrm {Lat}_k\) are disjoint, we may sum up these inequalities to obtain

*M*/

*N*] on both sides, and using the inequalities \(x/2 \le [x] \le x\) valid for \(x \ge 1\), it follows that

To obtain the upper inequality in (53), we consider the same function *G*(*x*, *y*) that was used in the proof for Theorem 1.1, and split into two cases as above.

*G*is the same function used in square case. With this, we obtain the inequality

*m*,

*M*and

*n*,

*N*, respectively, it follows that the vector \(b\in \ell _2^{(N,M)}\) which satisfies \(Z_{(N,M)}b(n,m)=G(n/N,m/M)\) is the same as in the above case, whence \(\beta (b,N,M) \le C \log M\). Now, since \(\alpha (b,N,M)=\alpha ({\mathcal {F}}_{(N,M)}b,M,N)\), the relation (50) implies that

### Remark 7.5

The quantitative Balian–Low theorem in finite dimensions over finite rectangular lattices (Theorem 1.9) may be proved as in the square case, with the only modification being putting \(\phi = S_{M} P_{N} \Phi \) and \(\psi = S_{N} P_{M} \Psi \).

## Footnotes

## References

- 1.Auslander, L., Gertner, I., Tolimieri, R.: The discrete Zak transform application to time-frequency analysis and synthesis of nonstationary signals. IEEE Trans. Signal Process.
**39**, 825–835 (1991)CrossRefGoogle Scholar - 2.Auslander, L., Gertner, I., Tolimieri, R.: The Finite Zak Transform and the Finite Fourier Transform. Radar and sonar, Part II, IMA Vol. Math. Appl., vol. 39, pp. 21–35. Springer, New York (1992)Google Scholar
- 3.Balian, R.: Un principe d’incertitude fort en théorie du signal ou en mécanique quantique. C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Unive. Sci. Terre
**292**(20), 1357–1362 (1981)Google Scholar - 4.Battle, G.: Heisenberg proof of the Balian–Low theorem. Lett. Math. Phys.
**15**(2), 175–177 (1988)MathSciNetCrossRefGoogle Scholar - 5.Benedetto, J.J., Czaja, W., Gadziński, P., Powell, A.M.: The Balian–Low theorem and regularity of Gabor systems. J. Geom. Anal.
**13**(2), 239–254 (2003)MathSciNetCrossRefzbMATHGoogle Scholar - 6.Benedetto, J.J., Czaja, W., Powell, A.M., Sterbenz, J.: An endpoint \((1,\infty )\) Balian–Low theorem. Math. Res. Lett.
**13**(2–3), 467–474 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 7.Benedetto, J.J., Heil, C., Walnut, D.F.: Differentiation and the Balian–Low theorem. J. Fourier Anal. Appl.
**1**(4), 355–402 (1995)MathSciNetCrossRefzbMATHGoogle Scholar - 8.Bourgain, J.: A remark on the uncertainty principle for hilbertian basis. J. Funct. Anal.
**79**(1), 136–143 (1988)MathSciNetCrossRefzbMATHGoogle Scholar - 9.Daubechies, I.: The wavelet transform, time-frequency localization and signal analysis. IEEE Trans. Inform. Theory
**36**(5), 961–1005 (1990)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Daubechies, I., Janssen, J.E.M.: Two theorems on lattice expansions. IEEE Trans. Inform. Theory
**39**(1), 3–6 (1993)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Folland, G.B.: Harmonic Analysis in Phase Space. Princeton University Press, Princeton (1989)zbMATHGoogle Scholar
- 12.Gautam, S.Z.: A critical-exponent Balian–Low theorem. Math. Res. Lett.
**15**(3), 471–483 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - 13.Gröchenig, K.: An uncertainty principle related to the Poisson summation formula. Stud. Math.
**121**(1), 87–104 (1996)MathSciNetCrossRefzbMATHGoogle Scholar - 14.Gröchenig, K.: Foundations of Time-Frequency Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser Boston Inc., Boston, MA (2001)CrossRefzbMATHGoogle Scholar
- 15.Ghobber, S., Jaming, P.: On uncertainty principles in the finite dimensional setting. Linear Algebra Appl.
**435**(4), 751–768 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 16.Gröchenig, K., Malinnikova, E.: Phase space localisation of Riesz bases for \(\ell ^2({\mathbf{R}}^d)\). Rev. Mat. Iberoam.
**29**(1), 115–134 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 17.Heil, C., Powell, A.M.: Gabor Schauder bases and the Balian–Low theorem. J. Math. Phys.
**47**(11), 1–21 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 18.Heil, C.E., Powell, A.M.: Regularity for complete and minimal Gabor systems on a lattice. Ill. J. Math.
**53**(4), 1077–1094 (2009)MathSciNetCrossRefzbMATHGoogle Scholar - 19.Heil, C.E.: Wiener Amalgam Spaces in Generalized Harmonic Analysis and Wavelet Theory. ProQuest LLC, Ann Arbor, MI, Thesis (Ph.D.)–University of Maryland, College Park (1990)Google Scholar
- 20.Lammers, M., Stampe, S.: The finite Balian–Low conjecture. In: International Conference on Sampling Theory and Applications (SampTA), pp. 139–143 (2015)Google Scholar
- 21.Low, F.E.: Complete sets of wave packets. In: DeTar, C., et al. (eds.) A Passion for Physics: Essays in Honor of Geoffrey Chew, pp. 17–22. World Scientific, Singapore (1985)Google Scholar
- 22.Nazarov, F.L.: Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type. Algebra i Analiz
**5**(4), 3–66 (1993)MathSciNetzbMATHGoogle Scholar - 23.Nitzan, S., Olsen, J.-F.: From exact systems to Riesz bases in the Balian–Low theorem. J. Fourier Anal. Appl.
**17**(4), 567–603 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 24.Nitzan, S., Olsen, J.-F.: A quantitative Balian–Low theorem. J. Fourier Anal. Appl.
**19**(5), 1078–1092 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.