Inhomogeneous Poisson processes in the disk and interpolation

TL;DR

This paper explores the geometrical properties of inhomogeneous Poisson processes on the disk, characterizing conditions for sequences to be Carleson-Newman or unions of separated sequences, with implications for function space interpolation.

Contribution

It provides new characterizations of inhomogeneous Poisson processes related to Carleson-Newman sequences and interpolation in Hardy, Bloch, and Dirichlet spaces.

Findings

Characterization of Poisson processes as Carleson-Newman sequences

Conditions for processes to be unions of separated sequences

Discussion of measures for interpolation in function spaces

Abstract

We investigate different geometrical properties of the inhomogeneous Poisson point process $\Lambda_{\mu}$ associated to a positive, locally finite, $\sigma$-finite measure $\mu$ on the unit disk. In particular, we characterize the processes $\Lambda_{\mu}$ such that almost surely: 1) $\Lambda_{\mu}$ is a Carleson-Newman sequence; 2) $\Lambda_{\mu}$ is the union of a given number M of separated sequences. We use these results to discuss the measures $\mu$ such that the associated process $\Lambda_{\mu}$ is almost surely an interpolating sequence for the Hardy, Bloch or weighted Dirichlet spaces.