Abstract:
Consider a Gaussian Analytic Function on the disk. In joint work
with Yanqi Qiu and Alexander Shamov, we show that its zero set is a
uniqueness set for the Bergman space on the disk: in other words, almost
surely, there does not exist a nonzero square-integrable holomorphic
function with these zeros. The distribution of our random subset is
invariant under the group of isometries of the Lobachevsky plane; the
action of every hyperbolic or parabolic isometry is mixing. It follows,
in particular, that our set is neither sampling nor interpolating in the
sense of Seip. Nonetheless, in a sequel paper, joint with Yanqi Qiu, we
give an explicit procedure to recover a Bergman function from its values
on our set.
By the Peres and Virag Theorem, zeros of a Gaussian Analytic
Function on the disk are a determinantal point process governed by
the Bergman kernel, and we prove, for general determinantal point
processes, that reproducing kernels sampled along a trajectory form
a complete system in the ambient Hilbert space. The key step in
our argument is that the determinantal property is preserved under
conditioning.