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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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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

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