Phase Retrieval Of Finite Blaschke Projection In Hardy Space

Phase retrieval is a nonlinear problem that aims to reconstruct a target sig-nal,up to a unimodular scalar,from the intensity measurements.Phase retrieval by Fourier intensity measurements is a classical application in coherent diffraction imaging,and the modified Blaschke products(MBPs for short)are the generaliza-tion of classical linear Fourier atoms.Motivated by this,we firstly discuss phase retrieval of reconstructing the orthogonal projection ?(f)of signal f ?H2(D)onto Takenaka-Malmquist(TM for short)system from Blaschke intensity measurements.Specifically,to reconstruct P(f)?(?)(f,B{a0,a1,…,ak})B{a0,a1,…,ak} by the intensity measurements {|(f,B?k)|,|(f,B?k)|,|(f,B?k)|:k? 1},where f? H2(D)such that f(a0)?0,B{a0,a1,…,ak} and B?k,B?k,B?k are all the finite MBPs.For this,we establish the condition on {B?k,B?k,B?k:k?1} such that up to a unimodular scalar,P(f)can be determined with probability 1 by the above Blaschke intensity measurements.Moreover,a recursive reconstruction algorithm for the phase retrieval is established and numerical simulations are conducted to verify the result.Finally,a cyclic AFD phase retrieval algorithm is established for the best choice of {al} by Blaschke inten-sity measurements.Numerical experiments are conducted to verify that the approxi-mation of the orthogonal projection ?n(f)obtained by phase retrieval to signal f can be greatly improved under such a set of points.Its main contents are as follows:Firstly,we exactly express the partial fraction decomposition coefficient of MBP-s when ai,i?0,…,k are distinct.A sufficient condition is established for the deter-mination of a unique solution by a class of nonlinear equation system.Secondly,the orthogonal projection P(f)of f in Hardy space is reconstruct-ed by Blaschke intensity measurements.We investigate conditions on signal f?H2(D),point set {(?)and Blaschke product {B?k,B?k,B?k:k?1}.A recur-sive reconstruction algorithm is established and numerical simulations are conducted to verify our result.Thirdly,we establish a cyclic AFD phase retrieval algorithm.Numerical ex-periments show that by the above algorithm the approximation of the orthogonal projection Pn(f)obtained by phase retrieval to signal f can be improved.