  Generalizing previous work of Iwaniec, Luo, and Sarnak (2000), we use
information from one-level density theorems to estimate the proportion of
non-vanishing of $L$-functions in a family at a low-lying height on the
critical line (measured by the analytic conductor). To solve the Fourier
optimization problems that arise, we provide a unified framework based on the
theory of reproducing kernel Hilbert spaces of entire functions (there is one
such space associated to each symmetry type). Explicit expressions for the
reproducing kernels are given. We also revisit the problem of estimating the
height of the first low-lying zero in a family, considered by Hughes and
Rudnick (2003) and Bernard (2015). We solve the associated Fourier optimization
problem in this setting by establishing a connection to the theory of de
Branges spaces of entire functions and using the explicit reproducing kernels.
In an appendix, we study the related problem of determining the sharp
embeddings between the Hilbert spaces associated to the five symmetry types and
the classical Paley-Wiener space.
