%A Wei QU, Tao QIAN, Guantie DENG, Youfa LI, Chunxu ZHOU %T ANALYTIC PHASE RETRIEVAL BASED ON INTENSITY MEASUREMENTS %0 Journal Article %D 2021 %J Acta mathematica scientia,Series B %R 10.1007/s10473-021-0619-x %P 2123-2135 %V 41 %N 6 %U {http://121.43.60.238/sxwlxbB/CN/abstract/article_16586.shtml} %8 2021-12-25 %X 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 $|\langle f,E_{a_1\cdots a_n}\rangle|.$