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A class of iterative greedy algorithms related to Blaschke product

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Abstract

Möbius 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 defined by instantaneous frequencies of signals they represent. The 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 century. Of the positive frequency study a directly related topic is positive frequency decomposition of signals. The mainly focused methods of such decompositions include the maximal selection method and the Blaschke product unwinding method, and joint use of the mentioned methods. In this paper, we propose a class of iterative greedy algorithms based on the Blaschke product and adaptive Fourier decomposition. It 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 terms. Furthermore, we give a comprehensive and rigorous proof of the converging rate in terms of the zeros of the remainders. Finite Blaschke product methods are proposed to avoid the infinite phase derivative dilemma, and to avoid the computational difficulties.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 61471132 and 11671363) and the Science and Technology Development Fund, Macau Special Administration Region (Grant No. 0123/2018/A3). The first author and the second author thank Professors Ronald Coifman and Stefan Steinerberger for their beneficial discussions during their visits to Yale University in 2018 and their helpful suggestions for the improvement of this paper. The authors are grateful to the anonymous referees for their careful reading of the manuscript and many helpful suggestions.

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Authors and Affiliations

  1. Macao Institute of Systems Engineering, Macau University of Science and Technology, Macao, China

    Tao Qian

  2. School of Applied Mathematics, Guangdong University of Technology, Guangzhou, 510006, China

    Lihui Tan

  3. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China

    Jiecheng Chen

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  1. Tao Qian
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  2. Lihui Tan
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  3. Jiecheng Chen
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Correspondence to Lihui Tan.

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Qian, T., Tan, L. & Chen, J. A class of iterative greedy algorithms related to Blaschke product. Sci. China Math. 64, 2703–2718 (2021). https://doi.org/10.1007/s11425-020-1706-5

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  • DOI: https://doi.org/10.1007/s11425-020-1706-5

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