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 $deg(P)<deg(Q)\le M$. We prove that for any $\epsilon >0$ and any $M\in \mathbb{N}$, there exists a universal set $\hat{I}›\subset \mathbb{R}$ of upper Beurling density less than $1+\epsilon$ such that the system $$\left\{e^{2\ddot{I}€i\hat{I}»t}g(t-n):(\hat{I}»,n)\in \hat{I}›\times \mathbb{Z}\right\}$$ forms a frame in $L^2(\mathbb{R})$ for any well-behaved rational function $g$.

22 pages