论文标题:修正Bernstein-Kantorovich型算子的逼近问题
On Approximation Properties of Weighted Bernstein-Kantorovich Operators
论文作者
论文导师 杨文善,论文学位 硕士,论文专业 应用数学
论文单位 浙江师范大学,点击次数 103,论文页数 41页File Size1525K
论文网 http://www.lw23.com/lunwen_384649062/
K_n~* operator;; H_p~(α,β) space;; q-Bernstein operator;; approximation problem
本文对Bernstein-Kantorovich型算子进行了一些修正,构造了一类新型算子,并用该算子解决了一类加权可积函数的逼近问题,得到了一些结果。特别地,本文对该类新型算子本身的性质做了细致的研究,尤是其规范化因子的形式及在区间端点附近的性态。从中可以看出它对这类新型算子的逼近问题起着至关重要的作用。同时,对近年来受国内外学者广为关注的一个新型算子q-Bernstein算子的几个问题也进行了研究。 以下是论文概要,全文共分四个部分。 第一部分,介绍了Bernstein-Kantorovich型算子的研究背景及发展过程、现状,提出了一个新算子,并给出了涉及该问题的一系列记号及定义。 第二部分,对构造的新算子形式进行了探索,给出了其规范化因子的具体形式;同时,刻画了规范化因子的极值性,说明其性质与x的位置密切相关,尤其在区间端点附近的性态对该算子的逼近问题起着关键作用,并找到了它的最高阶。 第三部分,研究了该算子对一类端点附近积分发散函数的逼近问题。首先解决了其加权有界性,然后在新的范数定义下,根据其规范化因子的性质,将区间分成三部分进行讨论,最终利用K-泛函得到了该算子在新的函数空间下的逼近速度。 第四部分,q-Bernstein算子也是Bernstein算子的一个变形。本部分将给出参数q在0与1之间扰动时q-Bernstein算子的饱和性理论,同时通过精细计算估计出在参数q趋于无穷大时的收敛速度。
In the present thesis, modified Bernstein-Kantorovich operator through weighting is introduced, which pulls approximation problems about functions into a new form. This also means that the new operator solves the approximation problems of a kind of not integrable functions. Some results about this problem are obtained. The properties of this new operator are studied, especially the formula of the standardized factor and its behavior around the two endpoints which plays a. quite important role in the approximation problem. Meanwhile, this thesis is also concerned with the q-Bernstein operator which has stirred the interest of many people home and abroad these years. Following is the structure of this thesis. The first section introduces the background and the development actuality of the Bernstein-Kantorovich operator, as well as the definition of the modified Bernstein-Kantorovich operator. The necessary declarations of the definitions and marks are also given in this part. The second section studies the properties of the new operator. The formula of the standardized factor is presented. Furthermore, the extreme properties of this standardized factor are considered, which shows the close connection between the properties and the value of x together with the essentiality of the standardized factor behavior around the two endpoints to the approximation problem. At last of this part give the maximum order of the standardized factor. The third section studies the approximation problem of a kind of not integrable functions with this new operator. First, deal with the boundness of the operator with the weighting. Then divide up [0,1] into three parts based on the properties of the standardized factor under the new norm. At last, using K-functional estimate the approximation degree in the new functions space. The forth section show the saturation theorem of q-Bernstein operator with the parameter q varying between 0 and 1 and estimate the convergence rate as the parameter q→∞through the fine calculation.
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