Bernstein算子及其推广的逼近性质

发布时间：2019-01-26 11:32

【摘要】：由于其在构造上的简洁性,又能够保持目标函数的单调性、凸性等优良性质,Bernstein算子在算子逼近乃至整个函数逼近论中一直占有非常重要的地位.Bernstein算子在泛函分析、计算数学和学习理论等领域得到了广泛的应用.本文主要研究Bernstein算子及其重要推广形式-Bernstein-Stancu算子的逼近性质,主要内容可以概括如下：第一章.简要介绍Bernstein算子及其一些重要推广形式的已有研究结果,特别是一些和本文内容有较大关联的研究情况.第二章.考虑具有端点奇性加权连续函数空间Cω中修正B ernstein算子Bn*(f,x)的加Jacobi权的逼近问题,利用加权K-泛函和加权光滑模,得到该算子在Cω空间中加权逼近的Stechkin-Marchaud型不等式.第三章.建立前述修正Bernstein算子Bn*(f,x)在Cω空间中的算子导数与函数光滑性之间的等价刻画,将王([35])的结论从连续函数空间推广到Cω空间,并且去掉了权函数w(x)=Xα(1-x)b中对参数a,b的上界限制.第四章.考虑了一种加权Bernstein-Durrmeyer算子在加权Lωp空间中的逼近问题,建立了逼近的正、逆定理.我们的结论中所涉及的Jacobi权函数ω(x)=xα(1-x)b去掉了a,b1-1/p的限制,从而本质性地推广了张([43])等的结论.第五章.研究了Gadjiev等([17])引入的一种新的Bernstein-Stancu型算子Sn,α,β(f,x)的逼近性质Gadjiev等([17])的结论表明Sn,α,β(f,x)可以逼近[0,1]的某个真子区间An上的连续函数,而我们揭示了Sn,α,β(f,x)可以较好地逼近f0,1]上定义的连续函数,从而本质性地推广了Gadjiev等([17])的结果,进一步,我们得到Sn,α,β(f,x)对C[0,1]空间中函数逼近的融整体和点态估计为一体的正、逆逼近定理.我们还本质性地改进了Tasdelen等([28])有关Sn,α,β[τ](f,x)算子对C[0,1]4空间中函数逼近的逼近阶估计.第六章.考虑一种Kantorovich型Bernstein-Stancu算子Sn,α,β*(f,x)在L[0,1]p空间的逼近性质,得到其在L[0,1]p空间的正、逆逼近定理,将Icoz[19])的有关结论从连续函数空间推广到了Linp空间.
[Abstract]:Because of its simplicity in construction and its excellent properties such as monotonicity and convexity of objective functions, Bernstein operators have always played an important role in approximation of operators and even in the whole theory of approximation of functions. Bernstein operators play an important role in functional analysis. Computational mathematics and learning theory have been widely used. In this paper, we study the approximation properties of Bernstein operators and Bernstein-Stancu operators, an important extension of them. The main contents can be summarized as follows: chapter 1. This paper briefly introduces the existing research results of Bernstein operators and some important forms of generalization, especially some research results which are related to the content of this paper. Chapter 2 In this paper, we consider the approximation problem with Jacobi weights for the modified B ernstein operator Bn* (FX) in the space C 蠅 of weighted continuous functions with singularities of endpoints. The weighted K-Functionals and the weighted modulus of smoothness are used. The Stechkin-Marchaud type inequality for weighted approximation of the operator in C 蠅 space is obtained. Chapter 3 In this paper, we establish the equivalent characterizations between the derivative of the modified Bernstein operator Bn* (FX) and the smoothness of the function in C 蠅 space, and extend the conclusion of Wang ([35]) from the continuous function space to the C 蠅 space. In addition, the upper bound of the parameter aqb in the weight function w (x) = X 伪 (1-x) b is removed. Chapter IV In this paper, the approximation problem of weighted Bernstein-Durrmeyer operators in weighted L 蠅 p spaces is considered, and the positive and inverse theorems of approximation are established. The Jacobi weight function 蠅 (x) = x 伪 (1-x) b in our conclusion removes the limitation of ahmb 1-1 / p, thus generalizes the conclusion of Zhang ([43]) et al. Chapter 5 In this paper, we study the approximation properties of a new Bernstein-Stancu type operator Sn, 伪, 尾 (FX) introduced by Gadjiev et al. ([17]). The results of Gadjiev et al. ([17]) show that Sn, 伪, 尾 (f), X) A continuous function on a true subinterval An of [0 ~ 1] can be approximated, and we reveal that Sn, 伪, 尾 (FX) can better approximate the continuous function defined on f _ 0 ~ (1). In this paper, we generalize the results of Gadjiev et al. ([17]). Furthermore, we obtain the positive and inverse approximation theorems of Sn, 伪, 尾 (FX) for the approximation of functions in C [0 ~ 1] space. We also improve the approximation order estimation of Sn, 伪, 尾 [蟿] (fnx) operators by Tasdelen et al ([28]) on C [0] 1] 4 spaces. Chapter 6 In this paper, we consider the approximation properties of Sn, 伪, 尾 * (FX) of a Kantorovich type Bernstein-Stancu operator in L [0 ~ 1] p space, and obtain its positive and inverse approximation theorems in L [0 ~ 1] p space. The conclusion of Icoz [19] is extended from continuous function space to Linp space.
【学位授予单位】：杭州师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O174.41

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