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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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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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PHASE RETRIEVAL OF TIME-LIMITED SIGNALS

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Acta Mathematica Scientia 2010年第1期30卷 39-46页

作者：付应雄李落清 Hubei Key Laboratory of Applied Mathematics Hubei University Faculty of Mathematics and Computer Science Hubei University

The problem of reconstructing a signalφ（x） from its magnitude ｜φ（x）] isof considerable interest to engineers and physicists. This article concerns the problem of determining a time-limited signal f with period 2π when ｜f（eix）l is known for x∈[-π,π]. It is shown that the conditions ｜g（eix）｜ = ｜f（eix）｜ and ｜g（ci（x＋b）） -g（eix）｜ =f（ei（x＋b）） - f（eix）｜, b ≠ 27π, together imply that either g = wf or g = v f, where both w and v have period b. Furthermore, if b/2π is irrational then the functions w and v b is rational then w takes the form reduce to some constants c1 and c2, respectively; ifb/2π is rational then w takes the form w=elexB1（e1x）B2（elx）and v takes the form ei（x2πN/b＋a）B1（elx）B2（elx）,where B1 and B2 are blaschke products.

关键词： phase retrieval time-limited signal blaschke product Hardy space on theunit circle

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Reducing subspaces of multiplication operators on function spaces

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Applied Mathematics(A Journal of Chinese Universities) 2013年第4期28卷 395-404页

作者： GUO Kun-yu HUANG Han-song School of Mathematical Sciences Fudan University Department of Mathematics East China University of Science and Technology

This survey presents the brief history and recent development on commutants and reducing subspaces of multiplication operators on both the Hardy space and the Bergman space, and von Neumann algebras generated by multi... 详细信息

关键词： reducing subspace commutant von Neumann algebra blaschke product holomorphic covering map

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THE EXTREME POINTS OF SEVERAL CLASSES OFANALYTIC FUNCTIONS

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Acta Mathematica Scientia 1999年第4期19卷 457-462页

作者：彭志刚 College of Mathematics and Computer Science Hubei University Wuhan 430062 China

Let P-n(c(1),c(2),...,c(n-1)) = {p(z) : p(z) is analytic in \z\ 0 and p(z) = 1 + c(1)z + c(2)z(2) +...+ c(n-1)z(n-1) + d(n)z(n) +..., where c(1),c(2),...,c(n-1) are fixed complex constants}. Let P-R,P-n(b(1),b(2),...,b(n-1)) = {p(z) : p(z) is analytic in \z\ 0 and p(z) = 1 + b(1)z + b(2)z(2) +...+ b(n-1)z(n-1) + d(n)z(n) +..., where b(1),b(2),...,b(n-1) are fixed real constants and the coefficients of p(z) are real}. Let T-n(l(1),l(2),...,l(n-1)) = {f(z) : f(z) is analytic in \z\ < 1 and f(z) = z + l(1)z(2) + l(2)z(3) +...+ l(n-1)z(n) + d(n)z(n+1) +...; where l(1),l(2),...,l(n-1) are fixed real constants and the coefficients of f(z) are real}. It is understood that P-n(c(1),c(2),...,c(n-1)), P-R,P-n(b(1),b(2),...,b(n-1)) and T-n(l(1),l(2),...,l(n-1)) are not empty when the constants c(k)(k = 1,...,n-1), b(k)(k = 1,2,...,n-1) and l(k)(k = 1,...,n-1) satisfy certain conditions. This paper obtaines the extreme points of P-n(c(1),...,c(n-1)), P-R,P-n(b(1),...,b(n-1)) and T-n(l(1),...,l(n-1)).

关键词： blaschke product extreme point

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THE EXTREME POINTS OF A CLASS OF ANALYTIC FUNCTIONS WITH POSITIVE REAL PART AND A PRESCRIBED SET OF VALUES

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Acta Mathematica Scientia 2012年第5期32卷 1929-1936页

作者：彭志刚 Faculty of Mathematics and Computer Science Hubei University

Letζ =（0,z1,z2,···,zn） with ｜zj｜〈1for1≤j≤n,ω=（1,w1,w2,···,wn）,and P（ζ,ω） denote the set of functions p（z） that are analytic in D={z：｜z｜〈1} and satisfy Rep（z）〉0（｜z｜〈1）,p（0）=1,p（zj... 详细信息

关键词： extreme point positive real part blaschke product

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