Universal frame set for rational functions
Year: 2025
Type: preprint
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 $g$.
Source: arXiv (Cornell University)
Author: Semenov, Andrei V.
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7103160849
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