Complement property and frames in the phase retrieval problem

Sequences of elements from a Hilbert space with phase retrieval property are considered. We discuss a connection between the phase retrieval property, the complement property, and the full spark property (strongly complete systems). There are conditions on the reproducing kernels in the Hardy space of holomorphic functions which ensure phase recovery. We use not only Hilbert, but also Banach frames to construct such sequences of reproducing kernels.