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 reconstructi...

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Bibliographic Details
Published inGeometric and functional analysis Vol. 32; no. 2; pp. 135 - 192
Main Authors Bufetov, Alexander I., Qiu, Yanqi
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.04.2022
Springer Nature B.V
Subjects
Analysis
Harmonic functions
Hyperbolic coordinates
Hyperbolic functions
Mathematics
Mathematics and Statistics
Reconstruction
Point processes
Complex hyperbolic spaces
Weighted Bergman spaces
Patterson–Sullivan construction
32A36
Reconstruction of harmonic functions
Primary 60G55 Secondary 37D40
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Summary: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 .
ISSN:1016-443X
1420-8970
DOI:10.1007/s00039-022-00592-w