Reconstruction of Hp -Functions: Best Approximation,Regularization and Optimal Error Estimates

Let U be the open unit disc of the complex plane. We explicitly construct the best pointwise approximation for determining a function in the Hardy space H ^p (U) from measured data at a countable set of points in U whenever the data is exact. A regularization scheme is given to deal with the case of nonexact data. Moreover, optimal error estimates are also studied.     

Riemann zeta function and Lyapunov-type inequalities for certain higher order differential equations

This note generalizes the well known Lyapunov-type inequalities for second-order linear differential equations to certain 2M-th order linear differential equations with five types of boundary conditions. The usage of the best constant of some Sobolev-type inequalities clarify the process for obtaining such inequality and sharpen the result of Çakmak [2].     

The inverse eigenvalue problem of generalized reflexive matrices and its approximation

This paper studies inverse eigenvalue problems of generalized reflexive matrices and their optimal approximations. Necessary and sufficient conditions for the solvability of the problems are derived, the solutions and their optimal approximations are provided.     

Acoustic detection and classification of river boats

We present a robust algorithm to detect the arrival of a boat of a certain type when other background noises are present. It is done via the analysis of its acoustic signature against an existing database of recorded and processed acoustic signals. We characterize the signals by the distribution of their energies among blocks of wavelet packet coefficients. To derive the acoustic signature of the boat of interest, we use the Best Discriminant Basis method. The decision is made by combining the answers from the Linear Discriminant Analysis (LDA) classifier and from the Classification and Regression Trees (CART) that is also accompanied with an additional unit, called Aisles, that reduces false alarms rate. The proposed algorithm is a generic solution for process control that is based on a learning phase (training) followed by an automatic real time detection while minimizing the false alarms rate.     

Optimal uniform approximation on angles by entire functions

The paper discusses the best or optimal uniform approximation problem by entire functions on a closed angle Δ. This problem has been studied by M.V. Keldysch in [4], under the assumption that the functions ? subject to approximation are holomorphic in a larger angle containing Δ and there is no restriction on the growth of ? at infinity. In [8], the problem was investigated for a wider class of functions ? continuously complex differentiable on Δ, with sharper estimates on the growth of approximating entire functions, linked with the growth of ? on Δ and the differential properties of ? on the boundary of Δ. In this paper, we improve some of the results on entire approximation on angles, using new approximation ideas partially presented in [9] and [10].     

A Note on Pad ′e Approximants Pairs as Limits of Algebraic Polynomials Pairs of Weighted Best Approximation in Orlicz Spaces

In this short note, we show the behavior in Orlicz spaces of best approxima-tions by algebraic polynomials pairs on union of neighborhoods, when the measure of them tends to zero.     

基于漫反射光谱法下层组织信息的最佳探测位置的研究

由于人体组织结构的复杂性和个体差异性,采用光谱分析技术实现组织内部信息的检测方法中,仍存在着检测位置不明确的问题。采用多位置空间分辨漫反射光谱法对脂肪肌肉组织中内部信息的最佳探测位置进行了研究。由于目标组织为肌肉组织,为了消除由于不同人体皮下脂肪层厚度的差异以及目标层中多次后向散射光对测量结果引起的较大误差,并根据光在复杂生物组织中的传输模型,通过增加约束条件——两条理想的“香蕉型”路径—来定义有效光子比SNR,并以SNR作为评价最佳探测位置SDS_best的标准,对优化后的Monte Carlo仿真结果进行统计分析,分别研究了脂肪层厚度h_f、脂肪层吸收系数μ_af和肌肉层吸收系数μ_am与光源探测器间距SDS之间的关系,并以h_f为自变量与SDS_best建立线性回归模型。研究表明,(1)当0<h_f<0.6 cm时,μ_af和μ_am对SDS_best几乎没有影响,而h_f与SDS_best的建模相关系数R2为0.991 8;(2)任取脂肪层厚度值0.12与0.22 cm作预测,得到预测误差分别为0.030 14和0.020 16,预测误差均能控制在5%以内。该方法能更方便、快速地为测量组织内部信息确定最佳检测位置,并有效削弱浅表组织以及目标层中多次背向散射光的干扰。     

Best Meromorphic Approximation of Markov Functions on the Unit Circle

Let E \subset(-1,1) be a compact set, let μ be a positive Borel measure with support \supp μ =E , and let H _p (G), 1≤ p ≤∈fty, be the Hardy space of analytic functions on the open unit disk G with circumference Γ={z \colon |z|=1} . Let Δ _n,p be the error in best approximation of the Markov function \frac{1}{2π i} ∈t_E \frac{d μ(x)}{z-x} in the space L _p (Γ) by meromorphic functions that can be represented in the form h=P/Q , where P ∈ H _p (G), Q is a polynomial of degree at most n , Q\not \equiv 0 . We investigate the rate of decrease of Δ _n,p , 1≤ p ≤∈fty , and its connection with n -widths. The convergence of the best meromorphic approximants and the limiting distribution of poles of the best approximants are described in the case when 1<p≤∈fty and the measure μ with support E=[a,b] satisfies the Szegő condition ∈t_a^b \frac{\log(d μ/ d x)}{\sqrt{(x-a)(b-x)}} dx >- ∈fty. July 27, 2000. Final version received: May 19, 2001.     

Estimation of a parameter of Morgenstern type bivariate exponential distribution by ranked set sampling

Ranked set sampling is applicable whenever ranking of a set of sampling units can be done easily by a judgement method or based on the measurement of an auxiliary variable on the units selected. In this work, we consider ranked set sampling, in which ranking of units are done based on measurements made on an easily and exactly measurable auxiliary variable X which is correlated with the study variable Y. We then estimate the mean of the study variate Y by the BLUE based on the measurements made on the units of the ranked set sampling regarding the study variable Y, when (X ,Y) follows a Morgenstern type bivariate exponential distribution. We then consider unbalanced multistage ranked set sampling and estimate the mean of the study variate Y by the BLUE based on the observations made on the units of multistage ranked set sample regarding the study variable Y. Efficiency comparison is also made on all estimators considered in this work.