Details

Autor(en) / Beteiligte

• Li, You Fa
• Han, De Guang

Titel

Phase Retrieval of Real-valued Functions in Sobolev Space

Ist Teil von

• Acta mathematica Sinica. English series, 2018-12, Vol.34 (12), p.1778-1794

Ort / Verlag

Beijing: Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society

Erscheinungsjahr

2018

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• The Sobolev space H s (ℝ d ) 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 , k γ } ⊆ 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 (ℝ d ) can be determined, up to a global sign, by the phaseless measurements { | ⟨ f , ϕ ~ j , k γ ⟩ | } . 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 H s (ℝ d ) ∩ C 1 (ℝ d ), and prove that our iterated reconstruction procedure guarantees uniform convergence for any function f ∈ H s (ℝ d )∩ C 1 (ℝ d ) whose Fourier transform f ^ is L 1 -integrable. Moreover, numerical simulations are conducted to test the efficiency of the reconstruction algorithm.