[Paper Review] The Balian-Low theorem for locally compact abelian groups and vector bundles

Ulrik Enstad|arXiv (Cornell University)|May 16, 2019

Mathematical Analysis and Transform MethodsMathematics64 references5 citations

TL;DR

This paper establishes a topological obstruction to the existence of well-localized Gabor frames on locally compact abelian groups by linking the Balian–Low theorem to the nontriviality of a vector bundle over the compact quotient space (G/Λ) × (Ĝ/Λ⊥). Using Heisenberg modules and the Zak transform as a Hilbert C*-module isomorphism, it proves that the Balian–Low statement holds if and only if the associated vector bundle is nontrivial, and applies this to prove the theorem for G = R × Qp.

ABSTRACT

Let $\Lambda$ be a lattice in a second countable, locally compact abelian group $G$ with annihilator $\Lambda^{\perp} \subseteq \widehat{G}$. We investigate the validity of the following statement: For every $\eta$ in the Feichtinger algebra $S_0(G)$, the Gabor system $\{ M_{ au} T_{\lambda} \eta \}_{\lambda \in \Lambda, au \in \Lambda^{\perp}}$ is not a frame for $L^2(G)$. When $G = \mathbb{R}$ and $\Lambda = \alpha \mathbb{Z}$, this statement is a variant of the Balian-Low theorem. Extending a result of R. Balan, we show that whether the statement generalizes to $(G,\Lambda)$ is equivalent to the nontriviality of a certain vector bundle over the compact space $(G/\Lambda) imes (\widehat{G}/\Lambda^{\perp})$. We prove this equivalence using a connection between Gabor frames and 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 new Balian-Low theorem for the group $\mathbb{R} imes \mathbb{Q}_p$, where $\mathbb{Q}_p$ denotes the $p$-adic numbers.

Motivation & Objective

• To characterize when the Balian–Low theorem holds for locally compact abelian groups by relating it to topological invariants of time-frequency lattices.
• To extend the Balian–Low theorem beyond R^n by identifying the obstruction as the nontriviality of a vector bundle over (G/Λ) × (Ĝ/Λ⊥).
• To establish a new framework using Heisenberg modules and Hilbert C*-modules to reframe Gabor frame theory in noncommutative geometry.
• To prove the Balian–Low theorem for the group R × Qp, a non-archimedean example, using the bundle-theoretic criterion.
• To show that the Zak transform is an isomorphism of Hilbert C*-modules, identifying the associated vector bundle explicitly.

Proposed method

• Uses Heisenberg modules E∆(G) associated to lattices ∆ = Λ × Λ⊥ in G × Ĝ as Hilbert C*-modules over C*(∆) ≅ C(X), where X = (G/Λ) × (Ĝ/Λ⊥).
• Applies the Serre–Swan theorem to identify the module of continuous sections of a complex vector bundle EG,Λ over X.
• Shows that the Zak transform ZG,Λ is an isomorphism of Hilbert C*-modules: ZG,Λ : E∆(G) → Γ(EG,Λ).
• Reduces the Balian–Low statement to the question of whether EG,Λ is trivial: a singly generated module iff the bundle is trivial.
• Uses K-theory and Morita equivalence to relate the existence of a single generator to the topology of the bundle.
• Applies the criterion to the group G = R × Qp, proving the Balian–Low theorem holds there by showing EG,Λ is nontrivial.

Experimental results

Research questions

• RQ1 For which locally compact abelian groups G and lattices Λ ⊆ G does the Balian–Low theorem hold for the Gabor system G(η, Λ × Λ⊥)?
• RQ2 What is the topological obstruction to the existence of a well-localized Gabor frame with window in S₀(G) over a lattice Λ × Λ⊥?
• RQ3 How can the Zak transform be interpreted as an isomorphism of Hilbert C*-modules in the context of time-frequency analysis?
• RQ4 Is the Balian–Low theorem valid for non-archimedean groups such as R × Qp?
• RQ5 What is the relationship between the triviality of the vector bundle EG,Λ and the existence of a single generator for the Heisenberg module E∆(G)?

Key findings

• The Balian–Low statement holds for (G, Λ) if and only if the vector bundle EG,Λ over (G/Λ) × (Ĝ/Λ⊥) is nontrivial.
• The Zak transform implements a C*-module isomorphism between the Heisenberg module E∆(G) and the module of continuous sections of EG,Λ.
• The existence of a Gabor frame with window in S₀(G) over Λ × Λ⊥ is equivalent to the triviality of the bundle EG,Λ.
• The Balian–Low theorem is proven for the group G = R × Qp by showing that EG,Λ is nontrivial in this case.
• The result generalizes previous theorems by Kaniuth and Kutyniok, which required G to be compactly generated, by removing this assumption via bundle-theoretic methods.
• The paper establishes a new duality between Gabor frame theory and vector bundle topology via Heisenberg modules and the Zak transform.