Phase Retrieval Of Real-Valued Functions In Sobolev Space

Keywords

42C40; 65T60; 94A20; measurement function perturbation; phase retrieval; reconstruction stability; retrievable stability; Sobolev space

Abstract

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 {{ϕ~j,kγ}⊆H−s(Rd) to the phase retrieval problem for the real-valued functions in Hs(ℝd). We prove that any real-valued function f ∈ Hs(ℝd) can be determined, up to a global sign, by the phaseless measurements {|⟨f,ϕ~j,kγ⟩|}. 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.

Publication Date

12-1-2018

Publication Title

Acta Mathematica Sinica, English Series

Volume

34

Issue

12

Number of Pages

1778-1794

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1007/s10114-018-7422-1

Socpus ID

85048879132 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/85048879132

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