Abstract:
Consider a Gaussian Analytic Function on the disk, that is, a random series whose coefficients
are independent complex Gaussians. In joint work with Yanqi Qiu and Alexander Shamov, we show
that the zero set of a Gaussian Analytic Function 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 having these zeros. The key role in our argument is played by the determinantal structure
of the zeros, and we prove, in general, that the family of reproducing kernels along a realization of
a determinantal point process generates the whole ambient Hilbert space, thus settling a conjecture
of Lyons and Peres. In a sequel paper, joint with Yanqi Qiu, we study how to recover a holomorphic function from its values on our set. The talk is based on the preprints arXiv:1806.02306 and
arXiv:1612.06751
Contact us: