Acta mathematica scientia,Series B ›› 2021, Vol. 41 ›› Issue (6): 2123-2135.doi: 10.1007/s10473-021-0619-x

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Wei QU1, Tao QIAN2, Guantie DENG3, Youfa LI4, Chunxu ZHOU4   

  1. 1. Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China;
    2. Macau Center for Mathematical Sciences, Macau University of Science and Technolofy, China;
    3. School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China;
    4. College of Mathematics and Information Science, Guangxi University, Nanning 530004, China
  • Received:2021-05-10 Revised:2021-09-28 Online:2021-12-25 Published:2021-12-27
  • Supported by:
    Tao Qian was funded by The Science and Technology Development Fund, Macau SAR (File no. 0123/2018/A3). You-Fa Li was supported by the Natural Science Foundation of China (61961003, 61561006, 11501132), Natural Science Foundation of Guangxi (2016GXNSFAA380049) and the talent project of the Education Department of the Guangxi Government for one thousand Young-Middle-Aged backbone teachers. Wei Qu was supported by the Natural Science Foundation of China (12071035).

Abstract: This paper concerns the reconstruction of a function $f$ in the Hardy space of the unit disc $\mathbb{D}$ by using a sample value $f(a)$ and certain $n$-intensity measurements $|\langle f,E_{a_1\cdots a_n}\rangle|,$ where $a_1,\cdots,a_n\in \mathbb{D},$ and $E_{a_1\cdots a_n}$ is the $n$-th term of the Gram-Schmidt orthogonalization of the Szegökernels $k_{a_1},\cdots,k_{a_n},$ or their multiple forms. Three schemes are presented. The first two schemes each directly obtain all the function values $f(z).$ In the first one we use Nevanlinna's inner and outer function factorization which merely requires the $1$-intensity measurements equivalent to know the modulus $|f(z)|.$ In the second scheme we do not use deep complex analysis, but require some $2$- and $3$-intensity measurements. The third scheme, as an application of AFD, gives sparse representation of $f(z)$ converging quickly in the energy sense, depending on consecutively selected maximal $n$-intensity measurements $|\langle f,E_{a_1\cdots a_n}\rangle|.$

Key words: phase retrieval, Hardy space of the unit disc, Szegökernel, Takenaka-Malmquist system, Gram-Schmidt orthogonalization, adaptive Fourier decomposition

CLC Number: 

  • 30H10