Strict density inequalities for sampling and interpolation in weighted spaces of holomorphic functions

TL;DR

This paper establishes strict density inequalities for sampling and interpolation in weighted Fock spaces of entire functions, showing such spaces cannot have sets that are both sampling and interpolating, and constructs near-critical density sampling sets.

Contribution

It proves strict density inequalities for sampling and interpolation in weighted Fock spaces, answering Lindholm's question, and introduces new techniques combining complex analysis and frame theory.

Findings

Sampling sets with density arbitrarily close to the critical density are constructed.

Spaces do not admit sets that are both sampling and interpolating.

The results apply to several complex variables and involve novel methods from frame theory.

Abstract

Answering a question of Lindholm, we prove strict density inequalities for sampling and interpolation in Fock spaces of entire functions in several complex variables defined by a plurisubharmonic weight. In particular, these spaces do not admit a set that is simultaneously sampling and interpolating. To prove optimality of the density conditions, we construct sampling sets with a density arbitrarily close to the critical density. The techniques combine methods from several complex variables (estimates for $\bar \partial$) and the theory of localized frames in general reproducing kernel Hilbert spaces (with no analyticity assumed). The abstract results on Fekete points and deformation of frames may be of independent interest.