英语翻译1.IntroductionApproximationbypolynomialsistheoldestandsimplestwaytorepresentcomplicatedfunc-tionsdefinedoverfinitedomains.ThetheoryofapproximationbypolynomialswasstudiedandsolvedbyWeierstrassin1855:itispossib

　　英语翻译

　　1.Introduction

　　Approximationbypolynomialsistheoldestandsimplestwaytorepresentcomplicatedfunc-tionsdefinedoverfinitedomains.Thetheoryofapproximationbypolynomialswasstudied

　　andsolvedbyWeierstrassin1855:itispossibletoapproximateanyarbitrarycontinuous

　　functionf(x)byapolynomialandmaketheerrorlessthanagivenaccuracy²byincreasing

　　thedegreeoftheapproximatingpolynomial.BesidestheproofofWeierstrass,thereare

　　manyproofs,theonegivenbyLebesgueandtheproofofBernsteininwhichtheBernstein

　　polynomialswereintroducedaretwoexamples.Polynomialscanberepresentedinmany

　　differentbasessuchasthepower,Bernstein,Chebyshev,Hermite,andLegendrebasisforms.

　　TheBernsteinpolynomialsplayanimportantroleinCAGD,becausetheyarebasesofthe

　　Bernstein-B´ezierrepresentation.Sincethenatheoryofapproximationhasbeendeveloped

　　andmanyapproximationmethodshavebeenintroducedandanalyzed.Themethodofleast-squaresapproximationaccompaniedbyorthogonalpolynomialsisoneoftheseapproximation

　　methods.