The Patterson-Sullivan Interpolation of Pluriharmonic Functions for Determinantal Point Processes on Complex Hyperbolic Spaces

TL;DR

This paper explores the use of Patterson-Sullivan interpolation to recover and interpolate pluriharmonic and harmonic functions on complex hyperbolic spaces using determinantal point processes, establishing conditions for uniform and simultaneous interpolation.

Contribution

It provides new necessary and sufficient conditions for interpolation of pluriharmonic functions and demonstrates optimal and strong uniform interpolation results for weighted Bergman and Hardy spaces.

Findings

Patterson-Sullivan construction recovers Bergman functions from random samples

Necessary and sufficient conditions for pluriharmonic function interpolation

Impossibility results for uniform interpolation in certain spaces

Abstract

The Patterson-Sullivan construction is proved almost surely to recover a Bergman function from its values on a random discrete subset sampled with the determinantal point process induced by the Bergman kernel on the unit ball $\mathbb{D}_d$ in $\mathbb{C}^d$. For super-critical weighted Bergman spaces, the interpolation is uniform when the functions range over the unit ball of the weighted Bergman space. As main results, we obtain a necessary and sufficient condition for interpolation of a fixed pluriharmonic function in the complex hyperbolic space of arbitrary dimension (cf. Theorem 1.4 and Theorem 4.11); optimal simultaneous uniform interpolation for weighted Bergman spaces (cf. Theorem 1.8, Proposition 1.9 and Theorem 4.13); strong simultaneous uniform interpolation for weighted harmonic Hardy spaces (cf. Theorem 1.11 and Theorem 4.15); and establish the impossibility of the uniform simultaneous interpolation for the Bergman space $A^2(\mathbb{D}_d)$ on $\mathbb{D}_d$ (cf. Theorem 1.12 and Theorem 6.7).