The Patterson–Sullivan Reconstruction of Pluriharmonic Functions for Determinantal Point Processes on Complex Hyperbolic Spaces

The Patterson–Sullivan reconstruction 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 𝔹_d in ℂ^d . For supercritical weighted Bergman spaces, the reconstruction is uniform when the functions range over the unit ball of the weighted Bergman space. We obtain a necessary and sufficient condition for reconstruction of a fixed pluriharmonic function in the complex hyperbolic space of arbitrary dimension; prove simultaneous uniform reconstruction for weighted Bergman spaces as well as strong simultaneous uniform reconstruction for weighted harmonic Hardy spaces; and establish the impossibility of the uniform simultaneous reconstruction for the Bergman space A^2(𝔹_d) on 𝔹_d .