Phase and conjugate phase retrieval on Paley-Wiener spaced and CM

Phase retrieval is the recovery of unknown signals from measurements with noisy or lost phase. Recovery from loss of phase occurs in applications such as X-ray crystallography, optics, speech processing, and quantum information theory. In this thesis, we introduce the concept of conjugate phase retrieval in complex vector spaces and provide examples of real-valued vectors which allow conjugate phase retrieval but not phase retrieval. We completely characterize conjugate phase retrievable vectors in C2. In Paley-Wiener spaces, we exhibit a connection between sets of uniqueness and unsigned sampling. We prove that a set of uniqueness in PWq+Ci allows unsigned sampling in PW^, and provide examples suggesting the converse is true. We further show that complex phase retrieval is not possible using real-valued samples for functions in Paley-Wiener spaces.