Conjugate phase retrieval in shift-invariant spaces generated by a Gaussian

Conjugate phase retrieval considers the recovery of a function, up to a unimodular constant and conjugation, from its phaseless measurements. In this paper, we explore the conjugate phase retrieval in a shift-invariant space generated by a Gaussian funciton. First, we show that the modulus function in the Gaussian shift-invariant space can be determined from the phaseless Hermite samples taken on a discrete sampling set. We then show that a function in the shift-invariant space generated by a Gaussian can be uniquely determined, up to a unimodular constant and conjugation, from its phaseless Hermite samples on a discrete set. For the functions with finite coefficient sequences, we provide an explicit reconstruction procedure.