例一:求f(x)=e在0,上的三次最佳平方逼近多项式 解:显然利用勒让德正交多项式适合,因此 f(x)=ex,x∈[0,1<f(t)=e05e05,t∈[-1, 对f()展开。 10(t)=1,L1(t)=t,L2(t)=0.5(312-1),L3(1)=0.5(5t3-31) 00 2 33 b=(f,L0)=34366,b=(f,L1)=0.5634 h2=(f,L2)=0.0559,b3=(f,L3)=0.0019 p(t)=1.7183+0.8301+0.06985(312-1)+0.0033(5t3-31 p(x)=1.7183+0.8301(2x-1)+0.06985(3(2x-1)2-1) +0.0033(5(2x-1)3-3(2x-1)例一：求f (x) = e x在[0，1]上的三次最佳平方逼近多项式。 对 展开。 解：显然利用勒让德正交多项式适合，因此 ( ) ˆ ( ) , [ 1,1] ˆ ( ) , [0,1] 0.5 0.5 2 1 2 1 f t f x e x f t e e t t x t x =   =  − = + 0.0033(5(2 1) 3(2 1)). ( ) 1.7183 0.8301(2 1) 0.06985(3(2 1) 1) ( ) 1.7183 0.8301 0.06985(3 1) 0.0033(5 3 ) , ) 0.0019 ˆ , ) 0.0559 ( ˆ ( , ) 0.5634 ˆ , ) 3.4366, ( ˆ ( 7 2 , 5 2 , 3 2 2, ( ) 1, ( ) , ( ) 0.5(3 1), ( ) 0.5(5 3 ) 3 2 2 3 2 2 3 3 0 0 1 1 0 0 1 1 2 2 3 3 3 3 2 0 1 2 + − − −  = + − + − −  = + + − + − = = = = = = = =  = = = = = = = − = − x x p x x x p t t t t t b f L b f L b f L b f L a a a a L t L t t L t t L t t t ， 