求英语高手翻译：只要手译

Question

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.

Answer 1

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。

Answer 2

1。介绍
多项式逼近是最古老和最简单的方式来表示复杂的功能定义在有限域。的多项式逼近理论研究
并解决了在1855年由维尔斯特拉斯：它是可能的近似任意连续
由一个多项式的函数f（x），使错误小于一个给定的精度²通过增加
近似多项式的程度。除了维尔斯特拉斯证明，有
许多证据，一个由勒贝格和伯恩斯坦证明伯恩斯坦
多项式进行了介绍两个例子。多项式可以表示在许多
不同的基础，如电力，伯恩斯坦，切比雪夫，埃尔米特，和Legendre基础形式。
Bernstein多项式CAGD中发挥重要的作用，因为它们的基地
的Bernstein-B'ezier表示。从那时起近似的理论已被开发
许多近似方法已经介绍和分析。伴随着正交多项式的最小二乘逼近的方法是这些近似之一
方法。

Answer 3

1.介绍
多项式近似法是表征有限域上的复杂函数的最古老最简单的方法。多项式近似法的理论在1885年由
维尔斯特拉斯研究并解开：任意连续函数f(x)都可以由多项式作近似，通过增加近似多项式的次方数，可以将误差值控制在给定精度范围内。除了维尔斯特拉斯以外，还有很多人证明过，比如勒贝格和伯恩斯坦的证明就是很好的例子，其中伯恩斯坦多项式已经介绍过了。多项式的表征可以有很多基本型，比如幂，伯恩斯坦，切比雪夫，埃尔米特和勒让德的基本型。伯恩斯坦多项式在计算机辅助几何设计（CAGD）中起了非常重要的作用，因为它是伯恩斯坦-贝塞尔表征的基础。此后近似法得到了发展，引入和分析了很多多项式方法。最小平方近似法伴随正交多项式的方法就是近似法的其中一个例子。

数学好难。。。。我就翻了个意思，术语你自己再理理顺。