逼近定理

In chapter 4, a family of two-parameter operators on Abelian group is introduced and its interpolation theorem for approximation is established.

所得的结果包含了R.DeVore定理作为特款，还包含了算子逼近的插补定理，是非线性逼近方面的一项基础性工作。

To cousider the least,we have to establish these facts under some comparablystrong but not the most proper conditions.From [12] and [13],if we suppose thatfis Lipschitz continuous in t,then all the above conditions(H1)-(H3) canbe established to a class of approximating problems of (1)(2).So the followingtheorem can be proved by theorem 5 through approximation process.

这时，我们只好"退而求其次"，在较强的"看来不是最适当的条件"下，来建立这些事实，从[12]和[13]中我们看到，只要加上"〓对t为Lipschitz连续"，则对（1）（2）的逼近问题族，上述条件（H1）-（H3）都可建立起来，从而，通过逼近过程，用定理5就可得到如下的定理。

Finaly, we introduce the Bernstein-Kantorovich quasi interpolant and study the approximation equivalent theorem with pointwise modulus (o≤λ≤1 ) in space L[0,1] and approximation equivalent theorem with Ditzian-Totik modulusin space L_[0,1](1 ≤p ≤ω).

最后，引入Bernstein-Kantorovich拟中插式，利用点态光滑模ω_~(2r)(0≤λ≤1)讨论了其在L_∞[0，1]空间的逼近等价定理和其在L_p[0，1](1≤p≤∞)空间关于模ω_ψ~(2r)的逼近等价定理。

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。

Equidistance point and difference theory in theory of function approximation are studied. Meanwhile, the relation among difference, difference quotient and derivate is revealed. By drawing Lagrange's and Cauchy's theorem of mean on difference and Taylor's formula into difference function, four theorems, such as Lagrange's theorem of mean on difference, are concluded in simple way. On the basis of these conclusions, the asymptotic property of middle point is studied, a series of new conclusions are drawn and the discussions on the asymptotic property of middle point in differential mid-value are summarized.

对函数逼近论中等距节点和差分理论进行了研究，揭示了差分、差商与导数之间的联系；将Lagrange中值定理、Cauchy中值定理、Taylor公式引入到差分函数中，简明地推导出Lagrange差分中值定理等4个定理，并在此基础上对"中间点"的渐近性进行了研究，得出了一系列"中间点"的渐近性的结果，概括了有关文献对微分中值公式的"中间点"的渐近性的讨论；给出的引理改进了函数逼近论的证明方法，精简了函数逼近论中的一些内容。

We divide this interval into several subintervals, based on deviation points, find the best uniform approximation polynomial of each subinterval, and then denormalize the computation result to the final result.

论文摘要：根据切比雪夫定理，提出了一种适于FPGA实现的函数逼近新方法：通过区间映射关系，将函数定义域映射到较小的逼近区间内，并按照交错点对逼近区间进行分段，分别求出每段的最佳一致逼近多项式，对函数进行分段逼近，最后将逼近结果反映射，得到最终的函数计算结果。

In this dissertation, the best simultaneous approximation with weight to infinite sequence from simultaneous-suns in complex normed linear space is considered, and in the case when the weight satisfies a certain condition, the characterizations of approximation from a simultaneous-sun and uniqueness of approximation from a RS-set are obtained; moreover, in the case when the weight is free, the characterizations of approximation to a totally bounded sequence from a convex set are established.

本文在复赋范空间中研究了同时太阳集对无限序列的加权同时逼近问题，在权满足一定条件时，通过把所研究问题转化为连续向量值函数空间中相应集对一个上半连续函数的最佳逼近问题，得到了逼近的特征和RS集逼近的唯一性定理，有例表明它们是已有结果的本质推广；在权不受限制时，得到了凸集对全有界序列逼近的特征。

This project also obtained several limit theorems for some important dependent random variables and stochastic processes, such as the Strassen law of the iterated logarithm for negatively dependent random variables, strong limit theorems for mixing random vectors in Banach spaces, sample path properties for two-parameter fractional Wiener processes, and so on.

随机环境中的随机变量与随机过程的研究在国内外相当活跃，本项目主要研究它们的极限性质，着重研究了随机风景中随机变量与随机过程的极限性质，主要取得了以下几个结果：首先对简单对称的Kesten-Spitzer随机游动在低阶矩的条件下给出了强逼近，大大减弱了前人要求任意阶矩的条件，然后对独立风景中的一般随机变量给出了强逼近的一般性结果，由此导出在风景和随机变量都只具有低阶矩的条件下的独立但不同分布、混合相依变量的强逼近，在只有弱高于二阶矩的条件下得到了重相对数律和弱收敛；给出了连续时间参数的Brown风景中Brown运动和稳定风景中稳定过程的滞后增量和连续模等精确样本轨道性质；同时给出了一些重要的相依随机变量和过程的若干极限定理，如负相关随机变量的Strassen重对数律、抽象空间上混合相依变量的一些强极限定理成立的充分必要条件、两参数分数Wiener过程的样本轨道性质等。

At first, the existence of extended form of approach to continuous selection for any set valued mapping without any continuity restriction in para-compact metric space is proved ;by the topologically separated mappings , the approximate selection theorem of sub-lower-semi-continuous mapping is established, furthermore , continuous selection problem in H-space is studied. Next, with W-correspondence, an improved variational inequality is obtained ; by the H-KKM mapping ,Ky Fan\'s minimax inequality is generalized to H-space . At last, with H-convexity instead of the linear topological structure, a new version of Browder fixed point theorem is established.Chapter 3 deals with set-valued mapping vector variational inequality and minimax problems.

第二章首先在仿紧的度量空间上对任意的集值映射建立了新的逼近连续选择定理，利用映射的拓扑可分性，在H-空间上建立了次下半连续映射的逼近连续选择定理和一个新的连续选择定理；然后利用W-对应，在H-空间上建立了广义的变分不等式；利用H-KKM映射，在H-空间上建立了广义的KyFan极小极大不等式；最后，利用H-凸性代替拓扑线性结构，在H-空间上建立了一个新型的Browder不动点定理。

Equidistance point and difference theory in theory of function approximation are studied. Meanwhile, the relation among difference, difference quotient and derivate is revealed. By drawing Lagrange's and Cauchy's theorem of mean on difference and Taylor's formula into difference function, four theorems, such as Lagrange's theorem of mean on difference, are concluded in simple way. On the basis of these conclusions, the asymptotic property of middle point is studied, a series of new conclusions are drawn and the discussions on the asymptotic property of middle point in differential mid-value are summarized.

对函数逼近论中等距节点和差分理论进行了研究，揭示了差分、差商与导数之间的联系；将Lagrange中值定理、Cauchy中值定理、Taylor公式引入到差分函数中，简明地推导出Lagrange差分中值定理等4个定理，并在此基础上对&中间点&的渐近性进行了研究，得出了一系列&中间点&的渐近性的结果，概括了有关文献对微分中值公式的&中间点&的渐近性的讨论；给出的引理改进了函数逼近论的证明方法，精简了函数逼近论中的一些内容。

By the time of its fall, most of the prisoners were writers who had written against the corruptions of the government.

到它被攻陷的时候，里面多数的犯人是写了反对政府贪污文章的作家。

The most obvious variation to ovum morphological character was that the color was changed from light green to sepiaceous in embryonic development, and all the ovums were almost hatched after 96h.

在胚胎发育过程中卵的形态特征最明显的变化是颜色从淡绿到深褐色，卵在发育96h后卵基本全部孵化。