Sampling, interpolation and Riesz bases in small Fock spaces

TL;DR

This paper characterizes Riesz bases of reproducing kernels in small Fock spaces, linking them to interpolation sequences and densities, and extends classical results like Kadets–Ingham to this setting.

Contribution

It provides a complete description of Riesz bases in small Fock spaces and establishes a connection with interpolating sequences, extending classical theorems.

Findings

Characterization of Riesz bases in small Fock spaces.

Connection between Riesz bases and interpolating sequences.

Density conditions for interpolation and sampling.

Abstract

We give a complete description of Riesz bases of reproducing kernels in small Fock spaces. This characterization is in the spirit of the well known Kadets--Ingham 1/4 theorem for Paley--Wiener spaces. Contrarily to the situation in Paley--Wiener spaces, a link can be established between Riesz bases in the Hilbert case and corresponding complete interpolating sequences in small Fock spaces with associated uniform norm. These results allow to show that if a sequence has a density stricly different from the critical one then either it can be completed or reduced to a complete interpolating sequence. In particular, this allows to give necessary and sufficient conditions for interpolation or sampling in terms of densities.