Download or read book Applications of Complex Analysis to the Phase Retrieval Problem written by Rolando III. Perez and published by . This book was released on 2021 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The study of phase retrieval involves the recovery of a function f in some functionspace from given data about the magnitude of |f| (phaseless information) and other assumptions on f, where these other assumptions can be in terms of some transform of f. Phase retrieval problems are widely studied because of their physical applications in fields of science and engineering.In this thesis, our central objective is to apply complex analytic tools to determine the solutions and investigate the stability of certain phase retrieval problems. Firstly, we solve the phase retrieval problem for wide-band signals, which are functions with mildly decreasing Fourier transforms. To do so, we first translate the problem to functions in the Hardy spaces on the disc via a conformal bijection, and take advantage of the inner-outer factorization. We also consider the same problem coupled with additional magnitude constraints, and determine if these constraints force uniqueness of the solution. Secondly, we extend some uniqueness results on the phase retrieval problem on the Hardy space on the disc to more general situations. More precisely, we show that certain holomorphic functions are uniquely determined by their moduli on two intersecting segments or on two concentric circles. Finally, we investigate the effect of zero-flipping on the stability of the phase retrieval problem for functions in the Paley-Wiener class, where zero-flipping refers to the replacement of zeros by their complex conjugates. We represent zero-flipping as an operator, and use its Fourier analytic properties to show our stability results.