
ANALYTIC PHASE RETRIEVAL BASED ON INTENSITY MEASUREMENTS
Wei QU, Tao QIAN, Guantie DENG, Youfa LI, Chunxu ZHOU
Acta mathematica scientia,Series B
2021, 41 (6):
21232135.
DOI: 10.1007/s104730210619x
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 GramSchmidt 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 $\langle f,E_{a_1\cdots a_n}\rangle.$
Reference 
Related Articles 
Metrics

