作业帮求英语高手和数学高手来帮小弟翻译一下,谢了
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求英语高手和数学高手来帮小弟翻译一下,谢了
TRANSFORMATION OF CHEBYSHEV–BERNSTEIN POLYNOMIAL BASIS
Abstract — In paper [4], transformation matrices mapping the Legendre and Bern- stein forms of a polynomial of degree into each other are derived and examined. In this paper, we derive a matrix of transformation of Chebyshev polynomials of the first kind into Bernstein polynomials and vice versa. We also study the stability of these linear maps and show that the Chebyshev–Bernstein basis conversion is remarkably well-conditioned, allowing one to combine the superior least-squares performance of Chebyshev polynomials with the geometrical insight of the Bernstein form. We also compare it to other basis transformations such as Bernstein-Hermite, power-Hermite, and Bernstein–Legendre basis transformations.
Keywords: Bernstein polynomials, Chebyshev polynomials of first kind, least-squares approximation, orthogonal polynomials, basis transformation, condition number, per- turbation, computer aided geometric design.
1. Introduction
Approximation by polynomials is the oldest and simplest way to represent complicated func-tions defined over finite domains. The theory of approximation by polynomials was studied and solved by Weierstrass in 1855: it is possible to approximate any arbitrary continuous function f(x) by a polynomial and make the error less than a given accuracy by increasing the degree of the approximating polynomial . Besides the proof of Weierstrass, there are many proofs, the one given by Lebesgue and the proof of Bernstein in which the Bernstein
polynomials were introduced are two examples. Polynomials can be represented in many different bases such as the power, Bernstein, Chebyshev, Hermite, and Legendre basis forms. The Bernstein polynomials play an important role in CAGD, because they are bases of the Bernstein-B´ezier representation. Since then a theory of approximation has been developed and many approximation methods have been introduced and analyzed. The method of least- squares approximation accompanied by orthogonal polynomials is one of these approximation methods.
TRANSFORMATION OF CHEBYSHEV–BERNSTEIN POLYNOMIAL BASIS
关于多项式切比雪夫基和伯恩斯坦基的转换
可以是这样翻译吧？
TRANSFORMATION OF CHEBYSHEV–BERNSTEIN POLYNOMIAL BASIS
Abstract — In paper [4], transformation matrices mapping the Legendre and Bern- stein forms of a polynomial of degree into each other are derived and examined. In this paper, we derive a matrix of transformation of Chebyshev polynomials of the first kind into Bernstein polynomials and vice versa. We also study the stability of these linear maps and show that the Chebyshev–Bernstein basis conversion is remarkably well-conditioned, allowing one to combine the superior least-squares performance of Chebyshev polynomials with the geometrical insight of the Bernstein form. We also compare it to other basis transformations such as Bernstein-Hermite, power-Hermite, and Bernstein–Legendre basis transformations.
Keywords: Bernstein polynomials, Chebyshev polynomials of first kind, least-squares approximation, orthogonal polynomials, basis transformation, condition number, per- turbation, computer aided geometric design.
1. Introduction
Approximation by polynomials is the oldest and simplest way to represent complicated func-tions defined over finite domains. The theory of approximation by polynomials was studied and solved by Weierstrass in 1855: it is possible to approximate any arbitrary continuous function f(x) by a polynomial and make the error less than a given accuracy by increasing the degree of the approximating polynomial . Besides the proof of Weierstrass, there are many proofs, the one given by Lebesgue and the proof of Bernstein in which the Bernstein
polynomials were introduced are two examples. Polynomials can be represented in many different bases such as the power, Bernstein, Chebyshev, Hermite, and Legendre basis forms. The Bernstein polynomials play an important role in CAGD, because they are bases of the Bernstein-B´ezier representation. Since then a theory of approximation has been developed and many approximation methods have been introduced and analyzed. The method of least- squares approximation accompanied by orthogonal polynomials is one of these approximation methods.
TRANSFORMATION OF CHEBYSHEV–BERNSTEIN POLYNOMIAL BASIS
关于多项式切比雪夫基和伯恩斯坦基的转换
可以是这样翻译吧？
转换的切比雪夫伯恩斯坦多项式的基础
抽象——在纸[4],变换矩阵映射勒让德和伯尔尼-斯坦形式的一个多项式的程度成彼此派生和检查.在本文中,我们得到一个矩阵的变换的切比雪夫多项式的第一种成伯恩斯坦多项式,反之亦然.我们也研究稳定这些线性映射和表明,切比雪夫伯恩斯坦基础转换非常状态良好的,允许一个结合优越的最小二乘契比雪夫多项式的性能与几何的洞见伯恩斯坦形式.我们也比较它和其他基础转换如伯恩斯坦海曼,功率海曼,伯恩斯坦勒让德基础转换.
关键词:伯恩斯坦多项式,切比雪夫多项式的第一种,最小二乘逼近,正交多项式,根据转换、条件数、每- turbation,计算机辅助几何设计.
1.介绍
近似的多项式是最古老和最简单的方式来表示复杂函数对其定义在有限域.近似理论进行了研究和解决由多项式由维尔斯特拉斯在1855年:可以近似任意连续函数f(x)通过一个多项式和使误差小于给定精度提高的程度近似多项式.除了证明维尔斯特拉斯,有许多证明,给出了一个由勒贝格和证据的伯恩斯坦,伯恩斯坦
介绍了多项式就是两个例子.多项式可以用在许多不同的基地如电源,伯恩斯坦,切比雪夫,海曼,勒让德基础形式.伯恩斯坦多项式用CAGD中扮演着重要的角色,因为他们是基地的伯恩斯坦b´ezier表示.此后一个理论的近似已经开发和许多近似方法进行了介绍和分析.该方法最小二乘逼近,伴随着正交多项式是其中的一个近似方法.
抽象——在纸[4],变换矩阵映射勒让德和伯尔尼-斯坦形式的一个多项式的程度成彼此派生和检查.在本文中,我们得到一个矩阵的变换的切比雪夫多项式的第一种成伯恩斯坦多项式,反之亦然.我们也研究稳定这些线性映射和表明,切比雪夫伯恩斯坦基础转换非常状态良好的,允许一个结合优越的最小二乘契比雪夫多项式的性能与几何的洞见伯恩斯坦形式.我们也比较它和其他基础转换如伯恩斯坦海曼,功率海曼,伯恩斯坦勒让德基础转换.
关键词:伯恩斯坦多项式,切比雪夫多项式的第一种,最小二乘逼近,正交多项式,根据转换、条件数、每- turbation,计算机辅助几何设计.
1.介绍
近似的多项式是最古老和最简单的方式来表示复杂函数对其定义在有限域.近似理论进行了研究和解决由多项式由维尔斯特拉斯在1855年:可以近似任意连续函数f(x)通过一个多项式和使误差小于给定精度提高的程度近似多项式.除了证明维尔斯特拉斯,有许多证明,给出了一个由勒贝格和证据的伯恩斯坦,伯恩斯坦
介绍了多项式就是两个例子.多项式可以用在许多不同的基地如电源,伯恩斯坦,切比雪夫,海曼,勒让德基础形式.伯恩斯坦多项式用CAGD中扮演着重要的角色,因为他们是基地的伯恩斯坦b´ezier表示.此后一个理论的近似已经开发和许多近似方法进行了介绍和分析.该方法最小二乘逼近,伴随着正交多项式是其中的一个近似方法.