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.