THE ASYMPTOTIC FORMULA OF THE ORTHOGONAL RATIONAL FUNCTION ON THE UNIT CIRCLE

In this paper we obtained the asymptotic formula of the orthogonal rational function on the unit circle with respect to the weight function μ(z) with preasigned poles, which are in the exterior of the unit disk.

Mathematical theory of signal analysis vs. complex analysis method of harmonic analysis

We present recent work of harmonic and signal analysis based on the complex Hardy space approach.

Positive-instantaneous frequency and approximation

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 frequency expansion is a generalization of the Fourier series expansion.Mono-components are crucial to understand the concept instantaneous frequency.We will present several most important mono-component function classes.Decompositions of signals into mono-components are called adaptive Fourier decompositions(AFDs).Wc note that some scopes of the studies on the ID mono-components and AFDs can be extended to vector-valued or even matrix-valued signals defined on higher dimensional manifolds.We finally provide an account of related studies in pure and applied mathematics.

A class of iterative greedy algorithms related to Blaschke product

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 represent.The 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 century.Of the positive frequency study a directly related topic is positive frequency decomposition of signals.The mainly focused methods of such decompositions include the maximal selection method and the Blaschke product unwinding method,and joint use of the mentioned methods.In this paper,we propose a class of iterative greedy algorithms based on the Blaschke product and adaptive Fourier decomposition.It 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 terms.Furthermore,we give a comprehensive and rigorous proof of the converging rate in terms of the zeros of the remainders.Finite Blaschke product methods are proposed to avoid the in nite phase derivative dilemma,and to avoid the computational diculties.