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Gabor systems with Hermite functions of order n and oversampling greater than n+1 which are not frames

Markus Faulhuber

TL;DR

The paper demonstrates that the known sufficient density condition for Gabor frames with Hermite windows on lattices does not extend to non-uniform periodic index sets. By analyzing zeros of the Zak transform, it constructs explicit non-frame examples for Hermite orders $n=1,2,3$ with oversampling densities exceeding $n+1$ (e.g., $h_1$ at density $3$, $h_3$ at densities $5$ and $7$, and $h_2$ at density $5$), including non-lattice periodic configurations. The key idea is that if the Zak transform $\mathcal{Z} g$ has zeros at a chosen set of points, then the sum of squared magnitudes $\sum_m |\mathcal{Z}(\pi(z_m) g)|^2$ can vanish, forcing the lower frame bound to zero and destroying the frame property; this uses the Zak-diagonalization of the frame operator. The results illuminate the limitations of density-based criteria and duality principles in non-uniform time-frequency sampling and contribute to understanding the frame-set structure for Hermite windows and non-lattice configurations.

Abstract

We show that a sufficient density condition for Gabor systems with Hermite functions over lattices is not sufficient in general. This follows from a result on how zeros of the Zak transform determine the frame property of integer over-sampled Gabor systems.

Gabor systems with Hermite functions of order n and oversampling greater than n+1 which are not frames

TL;DR

The paper demonstrates that the known sufficient density condition for Gabor frames with Hermite windows on lattices does not extend to non-uniform periodic index sets. By analyzing zeros of the Zak transform, it constructs explicit non-frame examples for Hermite orders with oversampling densities exceeding (e.g., at density , at densities and , and at density ), including non-lattice periodic configurations. The key idea is that if the Zak transform has zeros at a chosen set of points, then the sum of squared magnitudes can vanish, forcing the lower frame bound to zero and destroying the frame property; this uses the Zak-diagonalization of the frame operator. The results illuminate the limitations of density-based criteria and duality principles in non-uniform time-frequency sampling and contribute to understanding the frame-set structure for Hermite windows and non-lattice configurations.

Abstract

We show that a sufficient density condition for Gabor systems with Hermite functions over lattices is not sufficient in general. This follows from a result on how zeros of the Zak transform determine the frame property of integer over-sampled Gabor systems.

Paper Structure

This paper contains 7 sections, 4 theorems, 52 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Hermite functions
  4. Gabor systems
  5. The Zak transform
  6. Results
  7. Open questions and new zeros

Key Result

Theorem 3.1

For $g \in M^1({\mathbb R})$, let $\{z_1, \ldots, z_M\} \subset [0,1)^2$ be distinct zeros of its Zak transform, i.e., $\mathcal{Z} g(z_m) = 0$, $m = 1, \ldots , M$. Then, the multi-window Gabor system $\mathfrak{G}(\{\pi(z_m)g\}_{m=1}^M,{\mathbb Z}^2)$ is not a frame for $L^2({\mathbb R})$. Equival

Figures (7)

  • Figure 1: Zeros of $\mathcal{Z} f(x,\omega)$ of a suitable even (left) and odd (right) function in $[0,1)^2$, obtained by parity. Further zeros may exist, depending on $f$.
  • Figure 2: The set $\Gamma = \bigcup_{m=1}^3 ({\mathbb Z}^2 +z_m)$, $z_m \in \{ (0,0), (\frac{1}{2},0), (0,\frac{1}{2})\}$ and its decomposition into three relatively shifted copies of the integer lattice ${\mathbb Z}^2$.
  • Figure 3: Known zeros of $\mathcal{Z}_{\sqrt{2}} h_{4\ell+3}$ in the fundamental cell of $\sqrt{2} {\mathbb Z} \times \frac{1}{\sqrt{2}} {\mathbb Z}$.
  • Figure 4:
  • Figure 5: Known zeros for $\mathcal{Z}_2 h_{4\ell+3}$ in the fundamental cell of $2 {\mathbb Z} \times \frac{1}{2} {\mathbb Z}$.
  • ...and 2 more figures

Theorems & Definitions (9)

  • Theorem 3.1
  • proof
  • Proposition 3.2
  • proof
  • Corollary 3.3
  • proof
  • Theorem 3.4
  • proof
  • Conjecture 4.1