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数学进展 - 2018, Vol. 47(3): 321-347
综述文章
一种新的基于非线性相位的Fourier理论及其应用
A Novel Fourier Theory on Non-linear Phases and Applications

钱涛
QIAN Tao

澳门大学科技学院数学系, 澳门, 3001
Faculty of Science and Technology, University of Macau, 3001, Macau, P. R. China

收稿日期: 2018-04-17
出版日期: 2018-06-08
2018, Vol. 47(3): 321-347
DOI: 10.11845/sxjz.2018002a


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摘要 信号的正频率表示自Fourier分析诞生以来一直都是物理学家、数学家以及信号分析工作者密切关注的问题. 基于调和分析和复分析方法,在过去近二十年里诞生了单分量函数理论以及基于单分量函数的函数(信号)表示理论.作为原创性理论这个方法将信号快速分解为一些具有正的非线性瞬时频率的基本信号之和.该理论植根于经典数学并可以推广到定义在高维流形上的向量值及矩阵值信号. 这从而也创立了高维空间中的有理逼近理论. 单分量函数理论包括正瞬时频率的数学定义及几个最重要的单分量函数类的刻画. 单分量函数的表示理论包括核心自适应 Fourier 分解(Core Adaptive Fourier Decomposition, 或 Core AFD) 及其若干变种,包括解绕 AFD, 循环 AFD, 再生核 Hilbert 空间的预—正交 AFD. 除了理论及方法的概述, 本文也给出了两个新证明: 迄今最一般的依据极大选择原理的自适应分解的收敛性的证明; 以及参数重复选择的及用到再生核导数的必要性的证明. 最后我们给出该理论与数学及信号分析中若干相关理论的联系, 以及该方法的某些应用.
关键词 Blaschke乘积单分量函数Hardy 空间内函数和外函数自适应Fourier分解Beurling-Lax定理再生核Hilbert空间    
Abstract:Ever since the time of Fourier, positive frequency representation of signals has always been a central problem for physicists, mathematicians and signal analysts. Based on harmonic analysis and complex analysis methods, a mono-component function theory and a function (signal) representation theory based on mono-component functions have been being established during the past two decades. Being an original theory, it rapidly decomposes a signal into a sum of certain basic signals with positive nonlinear instantaneous frequency. The theory has roots in classical mathematics and can be generalized to vector-valued and matrix-valued signals defined on high-dimensional manifolds. It also leads to establishment of rational approximation in higher dimensional spaces. The theory of mono-component functions includes the mathematical definition of positive instantaneous frequencies and description of several most important mono-component function classes. The representation theory of mono-component functions includes Core Adaptive Fourier Decomposition (or Core AFD) and several variants, including Unwinding AFD, Cyclic AFD, and Pre-orthogonal AFD in reproducing kernel Hilbert spaces. In addition to the overview of theory and methods, this paper also provides two new proofs: the most general proof so far of convergence of adaptive decomposition based on maximal selection principle; and a proof of the necessity of reproducing kernel derivatives for parameter repetition. Finally, we specify some connections between this new development with the existing pure and applied mathematics, as well as some information on applications.
Key wordsBlaschke product    mono-component    Hardy space    inner and outer functions    adaptive Fourier decomposition    Beurling-Lax theorem    reproducing kernel Hilbert space
PACS:  O1-0  
  O17  
  O23  
  O29  
基金资助:澳门大学Multi-Year Research Grant (MYRG) (No. MYRG2016-00053-FST)和澳门政府自然和科技基金 (No. FDCT 079/2016/A2).
通讯作者: E-mail: fsttq@umac.mo   
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