Phase Retrieval of Real-valued Functions in Sobolev Space  

作　　者：You Fa LI De Guang HAN 

机构地区：[1]College of Mathematics and Information Science, Guangxi University, Nanning 530004, P. R. China [2]Department of Mathematics, University of Central Florida, Orlando, FL 32816

出　　处：《Acta Mathematica Sinica,English Series》2018年第12期1778-1794,共17页数学学报（英文版）

基　　金：supported by Natural Science Foundation of China(Grant Nos.61561006 and11501132);Natural Science Foundation of Guangxi(Grant No.2016GXNSFAA380049);the support from NSF under the(Grant Nos.DMS-1403400 and DMS-1712602)

摘　　要：The Sobolev space HS（Rd） with s 〉 d/2 contains many important functions such as the bandlimited or rational ones. In this paper we propose a sequence of measurement functions { φj^r,k}∈C H^-S（R^d） to the phase retrieval problem for the real-valued functions in H^s（R^d）. We prove that any real-valued function f ∈ H^s （Rd） can be determined, up to a global sign, by the phaseless measurements {｜（ f, φj^r,k}｜}. It is known that phase retrieval is unstable in infinite dimensional spaces with respect to perturbations of the measurement functions. We examine a special type of perturbations that ensures the stability for the phase-retrieval problem for all the real-valued functions in Hs（Rd） ∩ C1（Rd）, and prove that our iterated reconstruction procedure guarantees uniform convergence for any function f ∈ Hs （Rd）∩ C1 （Rd） whose Fourier transform f is L1-integrable. Moreover, numerical simulations are conducted to test the efficiency of the reconstruction algorithm.The Sobolev space HS（Rd） with s 〉 d/2 contains many important functions such as the bandlimited or rational ones. In this paper we propose a sequence of measurement functions { φj^r,k}∈C H^-S（R^d） to the phase retrieval problem for the real-valued functions in H^s（R^d）. We prove that any real-valued function f ∈ H^s （Rd） can be determined, up to a global sign, by the phaseless measurements {｜（ f, φj^r,k}｜}. It is known that phase retrieval is unstable in infinite dimensional spaces with respect to perturbations of the measurement functions. We examine a special type of perturbations that ensures the stability for the phase-retrieval problem for all the real-valued functions in Hs（Rd） ∩ C1（Rd）, and prove that our iterated reconstruction procedure guarantees uniform convergence for any function f ∈ Hs （Rd）∩ C1 （Rd） whose Fourier transform f is L1-integrable. Moreover, numerical simulations are conducted to test the efficiency of the reconstruction algorithm.

关 键 词：Sobolev space phase retrieval measurement function perturbation retrievable stability reconstruction stability. 

分 类 号：O174[理学—数学]