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A class of iterative greedy algorithms related to Blaschke product

Science China Mathematics 2021年第12期64卷 2703-2718页

作者： Tao Qian Lihui Tan Jiecheng Chen Macao Institute of systems Engineering Macao University of Science and TechnologyMacaoChina School of Applied Mathematics Guangdong University of TechnologyGuangzhou 510006China Department of Mathematics Zhejiang Normal UniversityJinhua 321004China

Mobius transforms,Blaschke products and starlike functions as typical conformal mappings of one complex variable give rise to nonlinear phases with non-negative phase derivatives with the latter being de ned by instantaneous frequencies of signals they *** positive analytic phase derivative has been a widely interested subject among signal analysts(see Gabor(1946)).Research results of the positive analytic frequency and applications appears in the literature since the middle of the 20th *** the positive frequency study a directly related topic is positive frequency decomposition of *** mainly focused methods of such decompositions include the maximal selection method and the Blaschke product unwinding method,and joint use of the mentioned *** this paper,we propose a class of iterative greedy algorithms based on the Blaschke product and adaptive Fourier *** generalizes the Blaschke product unwinding method by subtracting constants other than the averages of the remaining functions,aiming at larger winding numbers,and subtracting n-Blaschke forms of the remaining functions,aiming at generating larger numbers of zero-crossings,to fast reduce energy of the remaining ***,we give a comprehensive and rigorous proof of the converging rate in terms of the zeros of the *** Blaschke product methods are proposed to avoid the in nite phase derivative dilemma,and to avoid the computational diculties.

关键词： complex Hardy space Mobius transform Blaschke product rational orthogonal system takenaka-malmquist system mono-component adaptive Fourier decomposition unwinding Blaschke expansion

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ANALYTIC PHASE RETRIEVAL BASED ON INTENSITY MEASUREMENTS

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Acta Mathematica Scientia 2021年第6期41卷 2123-2135页

作者： Wei QU Tao QIAN Guantie DENG Youfa LI Chunxu ZHOU Department of Applied Mathematics School of Mathematics and PhysicsUniversity of Science and Technology BeijingBeijing100083China Macao Center for Mathematical Sciences Macao University of Science and TechnolofyMacaoChina School of Mathematical Sciences Beijing Normal UniversityBeijing100875China College of Mathematics and Information Science Guangxi UniversityNanning530004China

This paper concerns the reconstruction of a function f in the Hardy space of the unit disc D by using a sample value f(a)and certain n-intensity measurements||,where a_(1)…a_(n)∈D,and E_(a1…an)is the n-th term of the Gram-Schmidt orthogonalization of the Szego kernels k_(a1),k_(an),or their multiple *** schemes are *** 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 *** 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||.

关键词： phase retrieval Hardy space of the unit disc Szegökernel takenaka-malmquist system Gram-Schmidt orthogonalization adaptive Fourier decomposition

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