英语翻译1.IntroductionApproximation by polynomials is the oldest

英语翻译
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 ² 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.

Introduction 介绍
Approximation by polynomials is the oldest and simplest way to represent complicated functions defined over finite domains.
由多项式表示的近似值是最古老和最简单的表示对做出定义的有限域的复杂函数.
The theory of approximation by polynomials was studied and solved by Weierstrass in 1855:
威乐尔斯特劳斯 (Weierstrass) 在1855年研究并解答了由多项式表示的近似值的原理：
it is possible to approximate any arbitrary continuous function f (x) by a polynomial and make the error less than a given accuracy ² by increasing the degree of the approximating polynomial .
用多项式约计任何任意连续函数 f(x) 和用增加近似多项式的次数比起已知精确度来很少产生误差,这一点可能的.
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.
除了威乐尔斯特劳斯的证明外还有许多证明,李博斯克 (Lebesgue) 作出的那个证明和伯恩斯坦 (Bernstein) 的证明就是两个例子,其中伯恩斯坦多项式曾被采用.
Polynomials can be represented in many different bases such as the power,Bernstein,Chebyshev,Hermite,and Legendre basis forms.
多项式可以用许多不同的基表示,诸如乘方、伯恩斯坦多项式、切比雪夫算法(Chebyshev)、厄米插值( Hermite) 和勒让德多项式 ( Legendre) 等基本形式.
The Bernstein polynomials play an important role in CAGD,because they are bases of the Bernstein-B’ezier representation.
伯恩斯坦多项式在计算机辅助几何设计 (CAGD) 中起着重要作用,因为它们是伯恩斯坦-布莱热表示法的基础.
Since then a theory of approximation has been developed and many approximation methods have been introduced and analyzed.
从此,近似值原理已经被发展起来,而且许多近似方法已经被采用并解析.
The method of least-squares approximation accompanied by orthogonal polynomials is one of these approximation methods.
与正交多项式同时存在的最小平方近似值的方法是这些近似法的其中之一.