Gabor orthonormal bases, tiling and periodicity

TL;DR

This work addresses the structure of Gabor orthonormal bases $\mathbf{G}(g,T,S)$ in $L^2(\mathbb{R})$ by developing a translational tiling viewpoint. It proves that when $g$ is compactly supported, both the time-shift set $T$ and the frequency-shift set $S$ must be periodic in dimension one, with a key tiling-based mechanism underpinning the result. The authors establish dual tiling identities $\sum_{t\in T}|g(x-t)|^2 = D(T)$ and $\sum_{s\in S}|\widehat{g}(\xi-s)|^2 = D(S)$, deducing positive finite densities and the relation $D(T)D(S)=1$, and then employ finite local complexity tiling theory to show the periodicity of $S$. The work further applies these methods to refine the Liu–Wang conjecture, deriving that $(\Omega,T)$ forms a tiling pair and $(\Omega,S)$ a spectral pair for $\Omega=\{x:g(x)\neq 0\}$, and offers a new proof of DL14 in the nonnegative window case. Overall, it advances understanding of the rigid structure of Gabor bases in one dimension and links tiling theory with time-frequency analysis.

Abstract

We show that if the Gabor system $\{ g(x-t) e^{2πi s x}\}$, $t \in T$, $s \in S$, is an orthonormal basis in $L^2(\mathbb{R})$ and if the window function $g$ is compactly supported, then both the time shift set $T$ and the frequency shift set $S$ must be periodic. To prove this we establish a necessary functional tiling type condition for Gabor orthonormal bases which may be of independent interest.

Theorems & Definitions (11)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• proof : Proof of Theorem \ref{['thmD1.4.3']}
• Theorem 2.1: KL96
• proof : Proof of Theorem \ref{['thmA3']}
• Theorem 3.1: KL96
• Theorem 3.2: IK13, KL16
• Conjecture 4.1: LW03
• Theorem 4.2: DL14