On the frame property of Hermite functions and exploration of their frame sets

TL;DR

The paper resolves a key threshold in the frame sets of Gabor systems with Hermite windows on square lattices: $\mathcal G(h_n,(1/\sqrt{n+1})\mathbb Z^2)$ is a frame for $L^2(\mathbb R)$ iff $n\ge 4$, clarifying the boundary of the safety region $ab<1/(n+1)$. The authors develop a boundary-analysis framework via the Janssen representation and detailed Laguerre-polynomial bounds, enabling tight control of the adjoint-lattice sums and enabling explicit verifications for $4\le n\le 32$ and uniform-n bounds for $n\ge33$. They also push beyond the safety region by handling specific orders: $n=15$ yields a frame for $\delta\ge11$ on the main diagonal, while $n=9$ provides a frame at density 3, illustrating complex structure of the frame set beyond traditional thresholds. The work highlights limitations of the Janssen test in small-n regimes and demonstrates how analytic and numerical bounds on Laguerre polynomials can extend our understanding of Hermite-frame sets, contributing to the broader program of decoding the mysterious frame sets of Hermite functions.

Abstract

We study Gabor frames with Hermite window functions. Gröchenig and Lyubarskii provided a sufficient density condition for their frame sets, which leads to what we call the "safety region". For rectangular lattices and Hermite windows of order 4 and higher, we enlarge this safety region by providing new points on the boundary of this region. For this purpose, we employ the Janssen representation of the frame operator to compare its distance to the identity in the operator norm. The calculations lead to estimates on series involving Laguerre polynomials with Gaussian weight functions.

Figures (8)

• Figure 1: For the $n$-th Hermite function we know that the Gabor system $\mathcal{G}(h_n,a {\mathbb Z} \times b {\mathbb Z})$ is a frame if $ab < 1/(n+1)$ and that it is not a frame if $a b \geq 1$. The frame set structure of $\mathfrak{F}(h_n)$, $n\geq 1$, between the "safety region" and the "no frame" region is currently only partially understood.
• Figure 2: Left: we split ${\mathbb Z}^2$ into a finite section (gray area), where we perform a finite number of computations, and a complementary region, where the series can be bounded by $0 < \varepsilon \ll 1$ sufficiently small. Right: Instead of splitting ${\mathbb Z}^2$ by means of the max-norm, we may use the Euclidean distance and split ${\mathbb Z}^2$ into layers $\ell_m({\mathbb Z}^2) = \{(k,l) \in {\mathbb Z}^2 \mid k^2+l^2 = m\}$. This also shows the connection to decomposing an integer into a sum of two squares.
• Figure 3: Values of the finite computations part of the Janssen test quantity for the Gabor systems $\mathcal{G}(h_n, (1/\sqrt{n+1}) {\mathbb Z}^2)$. The values for $n=1$ and $n=3$ are exact and the other values are given to 5 digits after the decimal point (rounded up).
• Figure 4: Values of the Janssen test quantity for $\mathcal{G}(h_n, 1/\sqrt{(n+1)} \ {\mathbb Z}^2)$ and $n \leq 120$.
• Figure 5: The functions $|\mathcal{L}_{15}(m \pi \delta)|e^{-m \pi \delta/2}$ for $m=1,2,4,5$, and $\delta\in[11,16]$.

Theorems & Definitions (28)

• Theorem : Gröchenig, Lyubarskii
• Theorem 1.1: Main result
• Definition 2.1: Modulation space
• Proposition 2.2: Fundamental identity of Gabor analysis
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5: Janssen representation
• Lemma 2.6: Janssen test
• proof
• Lemma 2.7