The Balian–Low theorem for locally compact abelian groups and vector bundles

Abstract

Let Λ be a lattice in a second countable, locally compact abelian group G with annihilator ${\mathrm{\Lambda }}^{\perp }\subseteq \hat{G}$. We investigate the validity of the following statement: For every η in the Feichtinger algebra $S_0(G)$, the Gabor system ${\{M_{\tau }{T}_{\lambda }\eta \}}_{\lambda \in \mathrm{\Lambda },\tau \in {\mathrm{\Lambda }}^{\perp }}$ is not a frame for $L^2(G)$. When Λ is a lattice in $G=\mathbb{R}$, this statement is a variant of the Balian–Low theorem. Extending a result of R. Balan, we show that whether the statement generalizes to $(G,\mathrm{\Lambda })$ is equivalent to the nontriviality of a certain vector bundle over the compact space $(G/\mathrm{\Lambda })\times (\hat{G}/{\mathrm{\Lambda }}^{\perp })$. We prove this equivalence using Heisenberg modules. More specifically, we show that the Zak transform can be viewed as an isomorphism of certain Hilbert $C^{⁎}$-modules. As an application, we prove a Balian–Low theorem in the new context of the group $\mathbb{R}\times {\mathbb{Q}}_p$, where ${\mathbb{Q}}_p$ denotes the p-adic numbers.

Introduction

In his work on projective modules over noncommutative tori [47], M. Rieffel introduced a class of Hilbert $C^{⁎}$-modules known as Heisenberg modules. These modules establish the Morita equivalence of twisted group $C^{⁎}$-algebras associated to a lattice Δ in the time-frequency plane of a locally compact abelian (LCA) group G. Heisenberg modules have been applied numerous times in operator algebras and noncommutative geometry, see for example [31], [32], [9], [8], [34].

It was shown by F. Luef in [36], [37] that the Morita equivalence of Heisenberg modules over noncommutative tori is closely related to the duality theory of regular Gabor frames. These connections were recently generalized to the setting of LCA groups [28]. Gabor frames are the objects of study in Gabor analysis, which can be considered a subfield of time-frequency analysis. The central problem of Gabor analysis is the recovery of signals from a discrete set of time-frequency translates of fixed functions in $L^2({\mathbb{R}}^n)$. While Gabor analysis is usually carried out in ${\mathbb{R}}^n$, much of the framework can be generalized to the setting of a LCA group G as follows: If $(x,\omega )$ is an element of the time-frequency plane $G\times \hat{G}$, the time-frequency shift operator $\pi (x,\omega )$ acts on functions $\xi \in L^2(G)$ via

$$\pi (x,\omega )\xi (t)=\omega (t)\xi (x^{-1}t)$$

for $t\in G$. By picking a lattice Δ in the time-frequency plane $G\times \hat{G}$ and a finite set of functions ${\eta }_1,\dots ,{\eta }_k\in L^2(G)$, one forms the associated multiwindow Gabor system as follows:

$$\mathcal{G}({\eta }_1,\dots ,{\eta }_k;\mathrm{\Delta })=\{\pi (z){\eta }_j:z\in \mathrm{\Delta },1\le j\le k\}.$$

To allow for stable reconstruction of functions in $L^2(G)$ from a multiwindow Gabor system, one requires the frame property due to Duffin and Schaeffer [10] to be satisfied. That is, if there exist constants $K,L>0$ such that

$$K{\Vert \xi \Vert }_2^2\le \sum _{j=1}^k\sum _{z\in \mathrm{\Delta }}|〈\xi ,\pi (z){\eta }_j〉{|}^2\le L{\Vert \xi \Vert }_2^2$$

for all $\xi \in L^2(G)$, one calls the multiwindow Gabor system $\mathcal{G}({\eta }_1,\dots ,{\eta }_k;\mathrm{\Delta })$ a multiwindow Gabor frame. In particular, a singlewindow Gabor system, or just Gabor system for short, is called a Gabor frame if it forms a frame for $L^2(G)$.

One of the main observations of [36], [28] is that if the windows ${\eta }_1,\dots ,{\eta }_k$ of a multiwindow Gabor frame over the lattice $\mathrm{\Delta }\subseteq G\times \hat{G}$ are well-localized, they can be interpreted as a set of generators for the Heisenberg module ${\mathcal{E}}_{\mathrm{\Delta }}(G)$ constructed from Δ. By well-localized, we mean that the generators are all elements of the Feichtinger algebra $S_0(G)$, a Banach space of test functions that is fundamental to time-frequency analysis. Since Heisenberg modules are finitely generated, an immediate consequence is the existence of a multiwindow Gabor frame $\mathcal{G}({\eta }_1,\dots ,{\eta }_k;\mathrm{\Delta })$ with ${\eta }_j\in S_0(G)$, $1\le j\le k$, for any given lattice Δ in $G\times \hat{G}$.

A longstanding problem in Gabor analysis on $G={\mathbb{R}}^n$ is whether one can find a (singlewindow) Gabor frame $\mathcal{G}(\eta ;\mathrm{\Delta })$ with a well-localized window $\eta \in S_0({\mathbb{R}}^n)$ over a given lattice Δ in the time-frequency plane ${\mathbb{R}}^n\times {\hat{\mathbb{R}}}^n\cong {\mathbb{R}}^{2n}$. One can ask the same question for a lattice Δ in the time-frequency plane $G\times \hat{G}$ of a locally compact abelian group G: Does there exist an $\eta \in S_0(G)$ for which $\mathcal{G}(\eta ,\mathrm{\Delta })$ is a Gabor frame? Or, in terms of Heisenberg modules, is the Heisenberg module ${\mathcal{E}}_{\mathrm{\Delta }}(G)$ singly generated? A basic restriction on the lattice is provided by one of the density theorems [22]: It is necessary that $\mathrm{vol}(\mathrm{\Delta })\le 1$ for Gabor frames $\mathcal{G}(\eta ,\mathrm{\Delta })$ with $\eta \in S_0(G)$ to exist [27, Theorem 5.6]. But in the group $G={\mathbb{R}}^n$, more is true:

Theorem 1.1

Let Δ be a lattice in the time-frequency plane ${\mathbb{R}}^n\times \widehat{{\mathbb{R}}^n}$ of ${\mathbb{R}}^n$. If there exists a function η in the Feichtinger algebra $S_0({\mathbb{R}}^n)$ for which $\mathcal{G}(\eta ,\mathit{\Delta })$ is a Gabor frame for $L^2({\mathbb{R}}^n)$, then $\mathrm{vol}(\mathit{\Delta })<1$.

The above result is an example of a Balian-Low theorem (BLT), as it is a non-existence result for well-localized Gabor frames at the so-called critical density $\mathrm{vol}(\mathrm{\Delta })=1$. The original Balian-Low theorem is due to R. Balian [4] and F. Low [35] and concerns lattices of the form $\mathrm{\Delta }=\alpha \mathbb{Z}\times \beta \mathbb{Z}$ for $\alpha ,\beta >0$ in $\mathbb{R}\times \hat{\mathbb{R}}$. It also uses a slightly more general notion of time-frequency localization that does not involve the Feichtinger algebra, but which is particular to $G=\mathbb{R}$. In [5], a proof of the original formulation of the Balian–Low theorem is deduced from the uncertainty principle, and a relation to noncommutative geometry is demonstrated in [38]. The amalgam Balian–Low theorem is another early version of the BLT that employs Wiener amalgam spaces [6], [24]. Versions of the Balian–Low theorem for more general lattices in ${\mathbb{R}}^n$ have since been proved, even for discrete sets Δ without any lattice structure [1]. Theorem 1.1 can also be deduced from the perturbation results of H. Feichtinger and N. Kaiblinger [15]. The converse of Theorem 1.1 remains open for general lattices, but was proved for the class of non-rational lattices in [28]. The proof uses Heisenberg modules and K-theory of noncommutative tori.

It is no coincidence that the setting of Theorem 1.1 is the group $G={\mathbb{R}}^n$. Indeed, Theorem 1.1 is easily shown to fail if one replaces ${\mathbb{R}}^n$ with an arbitrary LCA group G [19]. For instance, it fails for discrete or compact groups (see Proposition 5.7). One might then ask if there is a way to characterize the groups G for which Theorem 1.1 holds. This will be the setup in the present paper, but we restrict ourselves to the case where the lattice Δ takes the form $\mathrm{\Lambda }\times {\mathrm{\Lambda }}^{\perp }$, where Λ is a lattice in G and ${\mathrm{\Lambda }}^{\perp }$ is the annihilator of Λ in $\hat{G}$ (10). These lattices always have volume 1. We will consider the following statement in the setting $(G,\mathrm{\Lambda })$, where G is a second countable LCA group and Λ is a lattice in G.

Statement 1.2

For all $\eta \in S_0(G)$, the Gabor system

$$\mathcal{G}(\eta ,\mathit{\Lambda }\times {\mathit{\Lambda }}^{\perp })=\{\pi (\lambda ,\tau )\eta :\lambda \in \mathit{\Lambda },\tau \in {\mathit{\Lambda }}^{\perp }\}$$

is not a frame for $L^2(G)$.

Note that it is a consequence of Theorem 1.1 that Statement 1.2 holds true for any lattice Λ in $G={\mathbb{R}}^n$. The advantage of the formulation of Statement 1.2 is that the Zak transform can be employed. Originally introduced by I. Gelfand [17], the Zak transform was later generalized by A. Weil to the case of locally compact abelian groups [50]. It takes it name from the physicist J. Zak who discovered it independently [52]. Given a lattice Λ in the LCA group G, the Zak transform of a complex-valued function ξ on G is the function $Z_{G,\mathrm{\Lambda }}\xi :G\times \hat{G}\to \mathbb{C}$ given by

$$Z_{G,\mathrm{\Lambda }}\xi (x,\omega )=\sum _{\lambda \in \mathrm{\Lambda }}\xi (x\lambda )\omega (\lambda )$$

for $(x,\omega )\in G\times \hat{G}$. The above defines a continuous function if e.g. $\xi \in S_0(G)$.

The first proofs of the amalgam version of the BLT [6], [24] employed the Zak transform, and used the fact that it diagonalizes the frame operator associated to the Gabor system $\mathcal{G}(\eta ,\mathrm{\Lambda }\times {\mathrm{\Lambda }}^{\perp })$. In [29], E. Kaniuth and G. Kutyniok used the Zak transform to show that the Balian–Low statement in Statement 1.2 holds for all lattices in compactly generated, second countable, locally compact abelian groups with noncompact identity component. It is an open problem whether the hypothesis that G is compactly generated can be dropped from their theorem.

One of the main points we want to make in this paper is that the Zak transform has a natural interpretation as an isomorphism of Hilbert $C^{⁎}$-modules. When $\mathrm{\Delta }=\mathrm{\Lambda }\times {\mathrm{\Lambda }}^{\perp }$, the associated Heisenberg module ${\mathcal{E}}_{\mathrm{\Delta }}(G)$ becomes a Hilbert $C^{⁎}$-module over the (un-twisted) group $C^{⁎}$-algebra of Δ. Using the Fourier transform, this algebra can be identified as the continuous functions on the Pontryagin dual $X:=\hat{\mathrm{\Delta }}\cong (G/\mathrm{\Lambda })\times (\hat{G}/{\mathrm{\Lambda }}^{\perp })$. By the Serre–Swan theorem, the projective module ${\mathcal{E}}_{\mathrm{\Delta }}(G)$ must be isomorphic to a vector bundle over X. The role of the Zak transform in this respect is to identify the vector bundle in question. More precisely, we prove that there exists a complex vector bundle $E_{G,\mathrm{\Lambda }}$ for which the following holds:

Theorem A cf. Theorem 5.1 / Proposition 4.5

Let Λ be a lattice in the second countable LCA group G, and let Δ be the lattice $\mathit{\Lambda }\times {\mathit{\Lambda }}^{\perp }$ in $G\times \hat{G}$. Then the Zak transform implements an isomorphism of Hilbert $C^{⁎}$-modules

$$Z_{G,\mathit{\Lambda }}:{\mathcal{E}}_{\mathit{\Delta }}(G)\to \mathit{\Gamma }(E_{G,\mathit{\Lambda }}).$$

Here, ${\mathcal{E}}_{\mathit{\Delta }}(G)$ is the Heisenberg module associated to Δ, and $\mathit{\Gamma }(E_{G,\mathit{\Lambda }})$ is the module of continuous sections of the complex line bundle $E_{G,\mathit{\Lambda }}$ constructed in Section 4.2.

Now Statement 1.2 is equivalent to the nonexistence of a single generator for the Heisenberg module ${\mathcal{E}}_{\mathrm{\Delta }}(G)$. We show in Section 2.3 that the $C(X)$-module $\mathrm{\Gamma }(E)$ of continuous sections of a vector bundle E over a compact Hausdorff space is singly generated if and only if E is a trivial bundle. Consequently, we can formulate Statement 1.2 in terms of the vector bundle $E_{G,\mathrm{\Lambda }}$ from Theorem A as follows:

Theorem B cf. Theorem 5.4

Let Λ be a lattice in a second countable, locally compact abelian group G. Then the following are equivalent:

• $E_{G,\mathit{\Lambda }}$ is nontrivial.

• The Balian–Low statement (Statement 1.2) holds in this setting. That is, whenever $\eta \in S_0(G)$, then the Gabor system

  $$\mathcal{G}(\eta ,\mathit{\Lambda }\times {\mathit{\Lambda }}^{\perp })=\{\pi (\lambda ,\tau )\eta :\lambda \in \mathit{\Lambda },\tau \in {\mathit{\Lambda }}^{\perp }\}$$

  is not a frame for $L^2(G)$.

The above theorem builds upon an idea of R. Balan connecting Gabor superframes to vector bundles over the 2-torus ${\mathbb{T}}^2$ [3]. A special case of his result is that the amalgam version of the Balian–Low theorem is a consequence of the nontriviality of a certain line bundle over ${\mathbb{T}}^2$. If $G=\mathbb{R}$ and $\mathrm{\Lambda }=\alpha \mathbb{Z}$ in Theorem B, then the base space $X_{G,\mathrm{\Lambda }}$ is homeomorphic to ${\mathbb{T}}^2$, and we will indeed show that $E_{G,\mathrm{\Lambda }}$ in this case is closely related to Balan's bundle (cf. Example 4.6). Thus, Theorem B can be viewed as an extension of this special case of Balan's result to general second countable LCA groups.

We end this paper by applying Theorem B to prove that the Balian–Low statement (Statement 1.2) holds in a new setting: We set G to be the truncated adele group $\mathbb{R}\times {\mathbb{Q}}_p$ where ${\mathbb{Q}}_p$ denotes the p-adic numbers. It is a well-known fact from number theory that the group of p-adic rationals $\mathrm{\Lambda }=\mathbb{Z}[1/p]=\{a/p^k:a,k\in \mathbb{Z}\}$ embeds as a lattice in G. We then have the following:

Theorem C cf. Theorem 6.5

Let G be the group $\mathbb{R}\times {\mathbb{Q}}_p$, and let Λ be the lattice $\mathbb{Z}[1/p]$ embedded into G as in Section 6. Then The Balian–Low statement (Statement 1.2) holds for $(G,\mathit{\Lambda })$: That is, whenever $\eta \in S_0(\mathbb{R}\times {\mathbb{Q}}_p)$, the Gabor system

$$\mathcal{G}(\eta ,\mathit{\Lambda }\times {\mathit{\Lambda }}^{\perp })=\{(s,x)\mapsto e^{2\pi irs}{e}^{-2\pi i{\{rx\}}_p}\eta (s-q,x-q):q,r\in \mathbb{Z}[1/p]\}$$

is not a frame for $L^2(\mathbb{R}\times {\mathbb{Q}}_p)$.

The above theorem is the first Balian–Low theorem in the context of a LCA group which is not compactly generated. Hence, it is not covered by the result of Kaniuth and Kutyniok [29]. Number-theoretic groups such as $\mathbb{R}\times {\mathbb{Q}}_p$ and the full adeles over the rationals have not been explored much in Gabor analysis so far. In [11], examples of Gabor frames in these groups are constructed, and a mild Balian–Low type theorem, namely [11, Proposition 4.4], is also proved. However, the result only holds for functions in $S_0(\mathbb{R}\times {\mathbb{Q}}_p)$ of a very specific form, and Theorem C is a generalization to all generators in $S_0(\mathbb{R}\times {\mathbb{Q}}_p)$.

The text is structured as follows: In Section 2, we define Hilbert $C^{⁎}$-modules and their frames, and we discuss modules of sections of vector bundles. In Section 3, we introduce Gabor analysis on locally compact abelian groups and Heisenberg modules. In Section 4, we introduce the Zak transform, quasiperiodic functions and the vector bundle $E_{G,\mathrm{\Lambda }}$ from Theorem A. In Section 5, we show that the Zak transform gives an isomorphism of Hilbert $C^{⁎}$-modules, and prove theorems Theorem A and Theorem B. Then in Section 6, we prove a Balian–Low theorem for the group $\mathbb{R}\times {\mathbb{Q}}_p$, namely Theorem C. In the appendix, we have collected some basic results that are needed but do not constitute a part of the spirit of the main text.