Universal frame set for rational functions

TL;DR

The article constructs universal time-frequency sampling sets $\Lambda$ of near-critical density $D(\Lambda)\le 1+\varepsilon$ that yield Gabor frames $\mathcal{G}(g,\Lambda\times\mathbb{Z})$ for all well-behaved rational windows $g$ of fixed degree $M$. It develops a duality-based criterion linking the frame property to invertibility of a block-structured operator $L_{\xi}$ and uses a detailed determinant analysis of segment matrices to ensure invertibility for a.e. $\xi\in[0,1)$. The result is first established for simple poles (class $\mathcal{K}(N)$) in Theorem 1, and then extended to general rational windows in $\mathcal{K}_1(M)$ via a derivative/trick that reduces higher-order poles to combinations of simple-pole components, culminating in Theorem 2. This work advances non-classical frame theory by providing universal, near-density-optimal frames for broad rational windows, with potential implications for robust time-frequency representations and signal processing tasks.

Abstract

Let $g \in L^2(\mathbb{R})$ be a rational function of degree $M$, i.e. there exist polynomials $P, Q$ such that $g = {{P} \over {Q}}$ and $deg(P) < deg(Q) \leq M$. We prove that for any $\varepsilon>0$ and any $M \in \mathbb{N}$ there exists universal set $Λ\subset \mathbb{R}$ of density less than $1+\varepsilon$ such that the system $$\left\{ e^{2πi λt } g(t-n) \colon (λ, n) \in Λ\times \mathbb{Z} \right\}$$ is a frame in $L^2(\mathbb{R})$ for any well-behaved rational function $g$.

Theorems & Definitions (21)

• Definition 1.1
• Remark 1.2
• Definition 1.3
• Example 1.4
• Theorem 2.1
• proof
• Lemma 3.1
• Definition 3.2
• Lemma 5.1
• proof