正线性算子

In this paper we use pointwise modulus of smoothness to study approximation direct theorem and equivalent theorem for some linear operators and quasi-interpolant operators; Using pointwise modulus we discuss the strong converse inequality on K-functional; and using a modified weighted K-functional and weighted modulus of smoothness we study approximation with Jacobi weight on operator with non-zero first order moments.

本文利用点态光滑模ω_~(2r)来研究某些线性算子及逆中插式逼近正定理和等价定理；利用点态光滑模讨论其关于K-泛函的强逆不等式；同时利用一种改变的带权K-泛函和带权光滑模研究一阶矩不为零的算子的点态带Jacobi权逼近。

The study of direct and inverse theorems on the approximation of linear operators to functions in normed linear spaces is an important subject in the approximation theory. It is significant in theory and application.

线性算子对赋范线性空间中函数逼近正逆定理的研究是逼近论中重要的研究课题之一，在理论和实际应用上都具有重要的意义。

This dissertation first presents a simple introduction of some important basic problems ofoperator approximation theory,and history and research status of wavelet theory.

全文首先介绍了算子逼近理论的一些重要基本问题以及小波理论发展的历史和现状，指明了线性正多项式算子逼近收敛阶的受限性。

Based on general orthogonal polynomials and general hybrid orthogonal functions, a new linear, continuous and bounded operator has been proposed in the dissertation. The definitions and properties of PGOPO are given, a various operational rules of the operator have been set up in a systematic way, the convergence of approximate solutions has been discussed. After introducing PGOPO and discussing its main properties and operational rules, a systematic methodology for solving various control problems via PGOPO has been developed.

本文在广义正交多项式及广义混合正交函数的基础上，提出了一种新的线性、连续、有界算子—分段广义正交多项式算子，探讨了这种算子的基本性质，并系统地建立了与该算子有关的各种运算规则及收敛性结论，给出了它在控制理论中的各种应用。

In order to find a stable approximate solution of linear compact operator equation, the article introduces general theories about ill-posed problems, it bases on spectral theory of self-adjiont compact operators and the singular value decomposition for compact operators, avails singular system to give expression of the solution, and explains ill-posedness of compact operator equation roots in the property that the singular values trends to zero. Thereout, it is provided with theoretic support of building up regularization method by inducting regularization filter to weaken or filtrate the influence that the nature of the singular value being very close to zero has on the solutions stability.

为了得到线性紧算子方程稳定的近似解，介绍了不适定问题正则化的一般理论，以自伴紧算子的谱分析与紧算子奇异值分解为理论基础，利用奇异系给出了解的表达式，说明了紧算子方程不适定性的根源在于紧算子的奇异值趋于零的性质，由此通过引入正则化滤子函数来减弱或滤掉奇异值趋于零的性质对解的稳定性的影响，构造正则算子，从而提供了建立正则化方法的理论依据。

By using matrix norm, matrix singular value, non-negative matrix theory, the regions of numaerical range and spectum of operator polynomial and the regularity of operator linear pencil are obtained.

8应用矩阵范数理论、矩阵的奇异值理论、非负矩阵的理论、友矩阵、算子的谱理论为工具，获得了算子多项式数值域具有的特性及其包含范围，算子多项式数值域与n-次数值域的关系，最后研究了算子线性束正则性的等价条件及谱分布情况。

In chapter5, the serial properties of numerical range of operator polynomial are put firstly. Secondly, the relation between numerical range of operator polynomial and n-numerical range are considered. Thirdly, in the light of matrix norm, matrix singular value, the regions and bounds of numerical range and spectrum of operator polynomial are dicussed carefully.

第五章研究了算子多项式数值域的性质、算子多项式数值域与n-次数值域的关系，特别地利用矩阵范数、矩阵的奇异值、非负矩阵的理论、友矩阵为工具，给出了算子多项式数值域及谱的范围全面刻画，并深刻地研究了算子线性束的正则性的等价条件及谱分布。

The representation of all the solutions for the equation Lv=f is obtained through constructing a standard orthogonal basis of null space N.

通过构造零空间的一组标准正交基，得到了线性算子方程Lv=f的所有解的表达形式。

In this project, we study the theory of higher order differential equations in Banach spaces and related topics. We solve an open problem put forward by two American Mathematicians and two Italian Mathematicians concerning wave equations with generalized Weztzell boundary conditions, introduce an existence family of operators from a Banach space $Y$ to $X$ for the Cauchy problem for higher order differential equations in a Banach space $X$, establish a sufficient and necessary condition ensuring $ACP_n$ possesses an exponentially bounded existence family, as well as some basic results in a quite general setting about the existence and continuous dependence on initial data of the solutions of $ACP_n$ and $IACP_n$. We set up quite a few multiplicative and additive perturbation theorems for existence families governing a wide class of higher order differential equations, regularized cosine operator families, regularized semigroups, and solution operators of Volterra integral equations, obtain classical and strict solutions having optimal regularity for the inhomogeneous nonautonomous heat equations with generalized Wentzell boundary conditions, gain novel existence and uniqueness theorems,which extend essentially the existing results, for mild and classical solutions of nonlocal Cauchy problems for semilinear evolution equations, present a new theorem with regard to the boundary feedback stabilization of a hybrid system composed of a viscoelastic thin plate with one part of its edge clamped and the rest-free part attached to a visocelastic rigid body. Also we obtain many other research results.

在本研究中，我们对Banach空间中的高阶算子微分方程的理论以及相关理论进行了深入研究，解决了由美国和意大利的四位数学家联合提出的一个关于广义Wentzell边界条件下的波动方程适定性的公开问题，恰当地定义了Banach空间中的高阶算子微分方程Cauchy问题的算子存在族及唯一族，建立了齐次和非齐次高阶算子微分方程Cauchy问题适定性的判别定理，获得了关于高阶退化算子微分方程的算子存在族、正则余弦算子族、正则算子半群、Volterra积分方程解算子族的乘积扰动和混合扰动定理，得到了关于以依赖于时间的二阶微分算子为系数的一大类非自治热方程非齐次情形下的时变广义Wentzell动力边值问题的古典解、严格解的最大正则性结果，获得了半线性发展方程非局部Cauchy问题广义解和经典解存在唯一的判别条件，从实质上推广了现有的相关结果；得到了一部分边缘固定而另一部分附在一粘弹性刚体上的薄板构成的混合粘弹性系统的边界反馈稳定化的新稳定化定理，还建立了一系列其他研究结果。

The study of direct and inverse theorems on the approximation of linear operators to functions in normed linear spaces is an important subject in the approximation theory.

线性算子对赋范线性空间中函数逼近正逆定理的研究是逼近论中重要的研究课题之一，在理论和实际应用上都具有重要的意义。

Breath, muscle contraction of the buttocks; arch body, as far as possible to hold his head, right leg straight towards the ceiling (peg-leg knee in order to avoid muscle tension).

呼气，收缩臀部肌肉；拱起身体，尽量抬起头来，右腿伸直朝向天花板（膝微屈，以避免肌肉紧张）。

The cost of moving grain food products was unchanged from May, but year over year are up 8%.

粮食产品的运输费用与5月份相比没有变化，但却比去年同期高8％。