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Analytic Phase Retrieval Based on Intensity Measurements


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 \(\left| {\left\langle {f,{E_{{a_1} \cdots {a_n}}}} \right\rangle} \right|\), where a1, ⋯, \({a_n} \in \mathbb{D}\), and \({E_{{a_1} \cdots {a_n}}}\) is the n-th term of the Gram-Schmidt orthogonalization of the Szegö kernels \({k_{{a_1}}}, \cdots ,{k_{{a_n}}}\), 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 \(\left| {\left\langle {f,{E_{{a_1} \cdots {a_n}}}} \right\rangle} \right|\).

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Correspondence to Tao Qian.

Additional information

Dedicated to the memory of Professor Jiarong YU

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 (2016GXNS-FAA380049) 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).

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Qu, W., Qian, T., Deng, G. et al. Analytic Phase Retrieval Based on Intensity Measurements. Acta Math Sci 41, 2123–2135 (2021).

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Key words

  • phase retrieval
  • Hardy space of the unit disc
  • Szegö kernel
  • Takenaka-Malmquist system
  • Gram-Schmidt orthogonalization
  • adaptive Fourier decomposition

2010 MR Subject Classification

  • 30H10