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Phase Retrieval of Real-valued Functions in Sobolev Space

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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 {\(\{\tilde \phi _{j,k}^\gamma\} \subseteq H^{-s} (\mathbb{R}^d)\) 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 \(\{|\langle f, \tilde \phi _{j,k}^\gamma\rangle|\} \). 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 \(\hat f\) is L1-integrable. Moreover, numerical simulations are conducted to test the efficiency of the reconstruction algorithm.

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Acknowledgements

We appreciate the reviewer for the valuable suggestions that help to improve the presentation of the paper a lot.

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Authors and Affiliations

  1. College of Mathematics and Information Science, Guangxi University, Nanning, 530004, P. R. China

    You Fa Li

  2. Department of Mathematics, University of Central Florida, Orlando, FL, 32816, USA

    De Guang Han

Authors
  1. You Fa Li
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  2. De Guang Han
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Correspondence to You Fa Li.

Additional information

The first author is partially supported by Natural Science Foundation of China (Grant Nos. 61561006 and 11501132) and Natural Science Foundation of Guangxi (Grant No. 2016GXNSFAA380049); the second author acknowledges the support from NSF under the (Grant Nos. DMS-1403400 and DMS-1712602)

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Li, Y.F., Han, D.G. Phase Retrieval of Real-valued Functions in Sobolev Space. Acta. Math. Sin.-English Ser. 34, 1778–1794 (2018). https://doi.org/10.1007/s10114-018-7422-1

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

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