Circle packing and interpolation in Fock spaces

TL;DR

This paper improves the known geometric criteria for interpolation and sampling sequences in Fock spaces by leveraging circle packing results to refine the constants involved.

Contribution

It introduces sharper bounds for interpolation and sampling sequences in Fock spaces using circle packing techniques, improving previous results.

Findings

Improved constant for interpolation sequences in Fock spaces.

Derived similar bounds for sampling sequences.

Utilized circle packing results to refine geometric criteria.

Abstract

It was shown by James Tung in 2005 that if a sequence $Z=\{z_n\}$ of points in the complex plane satisfies $$\inf_{n\not=m}|z_n-z_m|>2/\sqrt\alpha,$$ then $Z$ is a sequence of interpolation for the Fock space $F^p_\alpha$. Using results from circle packing, we show that the constant above can be improved to $$\sqrt{2\pi/(\sqrt3\,\alpha)},$$ which is strictly smaller than $2/\sqrt\alpha$. A similar result will also be obtained for sampling sequences.