Universal frame set for rational functions

TL;DR

This paper constructs a universal set of frequency shifts that, combined with integer translations of any well-behaved rational function, forms a frame in L^2(R), with the set's density arbitrarily close to 1.

Contribution

It introduces a universal frequency set for rational functions of bounded degree, ensuring frame properties for all such functions with minimal density.

Findings

Universal set of frequency shifts exists for rational functions.

The set's density can be made arbitrarily close to 1.

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