Shift-invariant sampling in two-sided small Fock spaces
Yurii Belov, Mikhail Mironov
TL;DR
The paper addresses the problem of characterizing shift-invariant sampling sequences for the two-sided small Fock spaces $\mathcal{F}^p_\alpha$ over the full range $0<p\le\infty$. It develops a density- and Beurling balayage–based framework to describe when a discrete set $\Lambda$ remains sampling under all nonzero complex dilations, i.e., when $c\Lambda$ is sampling for every $c>0$. The key contributions are a complete density criterion for the $p=\infty$ case—$\Lambda$ is shift-invariant sampling iff the lower logarithmic density satisfies $D^-_{\log}(\Lambda)>2\alpha$—and a transfer principle that yields the same density threshold characterizing shift-invariant sampling for all $0<p<\infty$ once $\Lambda$ is expressible as a finite union of separated sequences with a separated subsequence meeting the density condition. The results are connected to Gabor frame theory, where Beurling-type tools and complete interpolating sequences underpin the sampling/interpolation correspondence, thereby advancing geometric descriptions in this non-Bargmann context. Overall, the work provides a sharp, density-based geometry for shift-invariant sampling in two-sided small Fock spaces and extends Beurling–Seip methods to the $0<p\le\infty$ regime, with implications for time-frequency analysis and frame theory.
Abstract
We consider the sampling problem for two-sided small Fock spaces $\mathcal{F}^p_α$, for the full range $0 < p \le \infty$. We establish a geometric description of shift-invariant sampling sequences, i.e., sequences $Λ$ such that $c Λ$ is sampling for all $c \in \mathbb{C} \setminus \{ 0 \}$.