Universal frame set for rational functions

Abstract

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