  We study the structure of the zeros of optimal polynomial approximants to
reciprocals of functions in Hilbert spaces of analytic functions in the unit
disk. In many instances, we find the minimum possible modulus of occurring
zeros via a nonlinear extremal problem associated with norms of Jacobi
matrices. We examine global properties of these zeros and prove Jentzsch-type
theorems describing where they accumulate. As a consequence, we obtain detailed
information regarding zeros of reproducing kernels in weighted spaces of
analytic functions.
