Universal frame set for rational functions

Summary · qwen2.5:32b

The study proves the existence of a universal set $\Lambda$ with an upper Beurling density less than $1+\varepsilon$ that generates a frame in $L^2(\mathbb{R})$ for any rational function $g$, highlighting its significance in constructing frames from rational functions without specific adjustments. This matters because it provides a generalized method to create frames for rational functions of bounded degree, enhancing the efficiency and applicability of frame theory in signal processing and harmonic analysis.

Excerpt

arXiv:2510.25930v2 Announce Type: replace 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 $\Lambda \subset \mathbb{R}$ of upper Beurling density less than $1+\varepsilon$ such that the system $$\left\{ e^{2\pi i \lambda t } g(t-n) \colon (\lambda, n) \in \Lambda \times \mathbb{Z} \right\}$$ forms a frame in $L^2(\mathbb{R})$ for any well-behaved rational function $g$.