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Intrinsic mono-component decomposition of functions: An advance of Fourier theory

MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2010年第7期33卷 880-891页

作者： Qian, Tao Univ Macau Dept Math Fac Sci & Technol Taipa Macao Peoples R China

We propose a function decomposition model, called intrinsic mono-component decomposition (IMD). It is a continuation of the recent study on adaptive decomposition of functions into mono-components (MCs). It is a further improvement of two recent results of which one is adaptive decomposition of functions into modified inner functions, and the other is decomposition by using adaptive Takenaka-Malmquist systems. The proposed new decomposition model is of less restriction and thus gains more adaptivity. The theory is valid to both the unit circle and the real line contexts. Copyright (C) 2009 John Wiley & Sons, Ltd.

关键词： complex hardy space inner and outer functions Nevanlinna factorization theorem Takenaka-Malmquist system analytic signal instantaneous frequency and amplitude mono-components Greedy algorithm adaptive Decomposition of functions

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Adaptive Decomposition by Weighted inner functions: A Generalization of Fourier Series

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JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS 2011年第2期17卷 175-190页

作者： Qian, Tao Tan, Li-Hui Wang, Yan-Bo Univ Macau Dept Math Fac Sci & Technol Taipa Macao Peoples R China Guangdong Univ Technol Sch Appl Math Guangzhou 510090 Guangdong Peoples R China

In recent study adaptive decomposition of functions into basic functions of analytic instantaneous frequencies has been sought. Fourier series is a particular case of such decomposition. Adaptivity addresses certain optimal property of the decomposition. The present paper presents a fast decomposition of functions in the L(2) (partial derivative D) spaces into a series of inner and weighted inner functions of increasing frequencies.

关键词： Fourier series inner and outer functions Hardy space The Nevanlinna factorization theorem Blaschke product Analytic signal Instantaneous frequency and amplitude Mono-components Adaptive decomposition of functions

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Positive-instantaneous frequency and approximation

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Frontiers of Mathematics in China 2022年第3期17卷 337-371页

作者： Tao QIAN Macao Center for Mathematical Science Macao University of Science and TechnologyMacaoChina

Positive-instantaneous frequency representation for transient signals has always been a great concern due to its theoretical and practical importance,although the involved concept itself is *** desire and practice of uniqueness of such frequency representation(decomposition)raise the related topics in *** approximately the last two decades there has formulated a signal decomposition and reconstruction method rooted in harmonic and complex analysis giving rise to the desired signal *** method decomposes any signal into a few basic signals that possess positive instantaneous *** theory has profound relations to classical mathematics and can be generalized to signals defined in higher dimensional manifolds with vector and matrix values,and in particular,promotes kernel approximation for multi-variate *** article mainly serves as a *** also gives two important technical proofs of which one for a general convergence result(Theorem 3.4),and the other for necessity of multiple kernel(Lemma 3.7).Expositorily,for a given real-valued signal f one can associate it with a Hardy space function F whose real part coincides with *** function F has the form F=f+iHf,where H stands for the Hilbert transformation of the *** develop fast converging expansions of F in orthogonal terms of the form F=∑k=1^(∞)c_(k)B_(k),where B_(k)'s are also Hardy space functions but with the additional properties B_(k)(t)=ρ_(k)(t)e^(iθ_(k)(t)),ρk≥0,θ′_(k)(t)≥0,*** original real-valued function f is accordingly expanded f=∑k=1^(∞)ρ_(k)(t)cosθ_(k)(t)which,besides the properties ofρ_(k)andθ_(k)given above,also satisfies H(ρ_(k)cosθ_(k))(t)ρ_(k)(t)sinρ_(k)(t).Real-valued functions f(t)=ρ(t)cosθ(t)that satisfy the conditionρ≥0,θ′(t)≥0,H(ρcosθ)(t)=ρ(t)sinθ(t)are called *** f is a mono-component,then the phase derivativeθ′(t)is defined to be instantaneous frequency of *** above described positive-instantaneou

关键词： Möbius transform blaschke product mono-component hilbert transform hardy space inner and outer functions adaptive fourier decomposition rational orthogonal system nevanlinna factorization beurling-lax theorem reproducing kernel hilbert space several complex variables Clifford alge-bra pre-orthogonal AFD

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Numerical stability of circular Hilbert transform and its application to signal decomposition

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APPLIED MATHEMATICS AND COMPUTATION 2019年 359卷 357-373页

作者： Sun, Xiaoyun Dang, Pei Macau Univ Sci & Technol Fac Informat Technol Macau Peoples R China

In this paper, we propose a novel method of computing Hilbert transform based on the mechanical quadrature method. Experiments show that the method outperforms the library function 'hilbert' in Matlab when the values of functions being handled are very large or approach to infinity, that is a problem we have to deal with when we compute, for instance, outer functions. As an application, we use the method to obtain the unwinding series of signals, which is a positive frequency decomposition of signals and is dependent of extraction of outer functions involving computation of Hilbert transform. The experimental results show better stability. Then, we give the transient time frequency distribution of the unwinding series of signals. Finally, experiments of noisy signals are given, and the results show that the introduced method can well resist some disturbance. (C) 2019 Elsevier Inc. All rights reserved.

关键词： Hilbert transform Mechanical quadrature Unwinding series inner and outer functions Transient time-frequency distribution

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ADAPTIVE DECOMPOSITION INTO MONO-COMPONENTS

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ADVANCES IN DATA SCIENCE AND ADAPTIVE ANALYSIS 2009年第4期1卷 703-709页

作者： Qian, Tao Wang, Yan-Bo Dang, Pei Univ Macau Dept Math Fac Sci & Technol Taipa Macao Peoples R China

This paper reviews some recent progress on adaptive signal decomposition into mono-components that are defined to be the signals of non-negative analytic phase derivatives.

关键词： Hilbert transform analytic signal analytic phase derivative mono-component Bedrosian identity adaptive decomposition starlike function p-starlike function inner and outer functions

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The commutant of analytic toeplitz operators

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Transactions of the American Mathematical Society 1973年 184卷 261-273页

作者： Deddens, James A. Wong, Tin Kin Department Of Mathematics University Of Kansas United States Department Of Mathematics Wayne State University United States

In this paper we study the commutant of an analytic Toeplitz operator* For cf> e H°°9 let cp — xF be its inner-outer factorization* Our main result is that if there exists X € C such that X factors as X = xi*2each X an inner function, and if F — X is divisible by each X,-, then \T$j =s j n\TpjThe key step in the proof is Lemma 2, which is a curious result about nilpotent operators* One corollary of our main result is that if X(z)= zn, n ^ 1, then\T1# p\\another is that if € H°° isunivalent then \T* =\TZ\ • We are also able to prove that if the inner factor of 4> is X(z) = zn, n £ 1, then \T functions of zs» We conclude by raising six uestions. © 1973 American Mathematical Society.

关键词： Analytic function inner and outer functions analytic Toeplitz operator pure is ometry commutant

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