Positive-instantaneous frequency and approximation  

作　　者：Tao QIAN 

机构地区：[1]Macao Center for Mathematical Science,Macao University of Science and Technology,Macao,China

出　　处：《Frontiers of Mathematics in China》2022年第3期337-371,共35页中国高等学校学术文摘·数学（英文）

基　　金：Macao University Multi-Year Research Grant(MYRG)MYRG2016-00053-FST;Macao Government Science and Technology Foundation FDCT 0123/2018/A3.

摘　　要：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 paradoxical.The desire and practice of uniqueness of such frequency representation(decomposition)raise the related topics in approximation.During 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 representations.The method decomposes any signal into a few basic signals that possess positive instantaneous frequencies.The 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 functions.This article mainly serves as a survey.It 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 f.Such function F has the form F=f+iHf,where H stands for the Hilbert transformation of the context.We 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,a.e.The 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 mono-components.If f is a mono-component,then the phase derivativeθ′(t)is defined to be instantaneous frequency of f.The above described positive-instantaneous fre

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 

分 类 号：O17[理学—数学]