A uniform approach to complete interpolating sequences for small Fock spaces with $p > 0$
Mikhail Mironov
TL;DR
This work delivers a uniform, perturbation-type characterization of complete interpolating sequences for both one-sided and two-sided small Fock spaces $\mathcal{F}^{p}_{\alpha+}$ and $\mathcal{F}^{p}_{\alpha}$ for all $0<p\le\infty$. The authors introduce a precise geometric framework based on $d_{\log}$ and $d_{\log_+}$ separations, and they reduce CIS to three concrete conditions involving a canonical perturbation $\delta_k$ in the exponential frequency representation, plus a bounded-average constraint. The core method combines canonical product estimates, sharp evaluation bounds, a canonical base sequence $\Gamma$, and a stability/Fredholm-style argument to transfer CIS from the base sequence to arbitrary CIS, with a bilateral extension revealing a curious p-periodicity: CIS for $\mathcal{F}^{p}_{\alpha}$ coincide for $p=1$, $p=2$, and $p=\infty$ but differ for other $p\ge1$. The results extend known CIS descriptions to the full range $0<p\le\infty$ and illuminate a deep connection between the geometry of sampling in small Fock spaces and perturbation theory in spectral sequences.
Abstract
We study complete interpolating sequences in two types of small Fock spaces, $\mathcal{F}^p_{α+}$ and $\mathcal{F}^p_α$, for $0 < p \le \infty$. One-sided small Fock spaces $\mathcal{F}^p_{α+}$ are well-studied spaces of entire functions with sub-exponential growth, while $\mathcal{F}^p_α$ are their two-sided analogue with a symmetric singularity at the origin. For one-sided small Fock spaces $\mathcal{F}^p_{α+}$, we provide a streamlined, perturbation-type description of complete interpolating sequences that unifies and extends earlier results for $1 \le p \le \infty$ to the full range $0 < p \le \infty$. For two-sided small Fock spaces $\mathcal{F}^p_α$, we establish a parallel characterization, revealing a curious periodicity phenomenon: complete interpolating sequences for $\mathcal{F}^p_α$ coincide exactly for $p = 1$, $p = 2$, and $p = \infty$, but differ for other $p \ge 1$.