. . .
kBn(f,x)??f()Pk(x)
nk?0其中Pk(x)???xk(1?x)n?k 当n?1时,
n?n??k??1?P0(x)???(1?x)
?0?P1(x)?x?B1(f,x)?f(0)P0(x)?f(1)P1(x)?1??????(1?x)sin(?0)?xsin22?0??x当n?3时,
?1?P0(x)???(1?x)3?0??1?22P(x)?1??x(1?x)?3x(1?x)?0??3?P2(x)???x2(1?x)?3x2(1?x)?1??3?3P3(x)???x?x3?3?
k?B3(f,x)??f()Pk(x)nk?0?0?3x(1?x)2sin?3?6?3x2(1?x)sin?3?x3sin?2
3332x(1?x)2?x(1?x)?x3225?33333?623?x?x?x222?1.5x?0.402x2?0.098x32. 当f(x)?x时,求证Bn(f,x)?x 证明:
若f(x)?x,则
kBn(f,x)??f()Pk(x)
nk?0.. ..
n. . .
k?n?k????x(1?x)n?kk?0n?k?n????nnkn(n?1)(n?k?1)kx(1?x)n?kk!k?0n(n?1)[(n?1)?(k?1)?1]kx(1?x)n?k(k?1)!k?1
n ??n?1?kn?k???x(1?x)k?1?k?1?n?n?1?k?1(n?1)?(k?1)?x??x(1?x)?k?1?k?1??x[x?(1?x)]n?1?x,xn线性无关
3.证明函数1,x,证明:
2若a0?a1x?a2x??anxn?0,?x?R
分别取x(k?0,1,2,k,n),对上式两端在[0,1]上作带权?(x)?1的积,得
??1????1??n?11??a?0n?1???a??0???01?????? ?????1???????an??0?2n?1?此方程组的系数矩阵为希尔伯特矩阵,对称正定非奇异, ?只有零解a=0。
?函数1,x,,xn线性无关。
?4。计算下列函数f(x)关于C[0,1]的f,f1与f2:
(1)f(x)?(x?1)3,x?[0,1]1(2)f(x)?x?,2
(3)f(x)?xm(1?x)n,m与n为正整数, (4)f(x)?(x?1)10e?x
解:
(1)若f(x)?(x?1)3,x?[0,1],则
.. ..
. . .
f?(x)?3(x?1)2?0
?f(x)?(x?1)3在(0,1)单调递增
f??maxf(x)0?x?1?max?f(0),f(1)? ?max?0,1??1f??maxf(x)0?x?1?max?f(0),f(1)? ?max?0,1??1f2?(?(1?x)dx)0161211712?[(1?x)]07
?771,x??0,1?,则 2(2)若f(x)?x?ff??maxf(x)?0?x?11012
??f(x)dx111?2?1(x?)dx221?4f
21
?(?f(x)dx)
0
1
2
12
121
?[?(x?)dx]2
023?6
(3)若f(x)?xm(1?x)n,m与n为正整数
当x??0,1?时,f(x)?0
.. ..
. . .
f?(x)?mxm?1(1?x)n?xmn(1?x)n?1(?1)?xm?1(1?x)n?1m(1?n?mmx)当x?(0,mn?m)时,f?(x)?0 ?f(x)在(0,mn?m)单调递减
当x?(mn?m,1)时,f?(x)?0
?f(x)在(mn?m,1)单调递减。
x?(
m
n?m
,1)f?(x)?0f??max0?x?1
f(x)?
?max??m? ?f(0),f(n?m)?
?
?
mmnn(m?n)m?n
f11??0f(x)dx??10xm(1?x)ndx???2(sin2t)m0(1?sin2t)ndsin2t???2sin2mtcos2n0tcost2sintdt?n!m!(n?m?1)!f2?[?112m2n20x(1?x)dx]?1?[?24m4ntd(sin2t)]20sintcos?1
?[?2202sin4m?1tcos4n?1tdt]?(2n)!(2m)![2(n?m)?1]!(4)若f(x)?(x?1)10e?x
当x??0,1?时,f(x)?0
.. ..
. . .
f?(x)?10(x?1)9e?x?(x?1)10(?e?x)?(x?1)9e?x(9?x)?0?f(x)在[0,1]单调递减。
f??maxf(x)?0?x?1?max?f(0),f(1)?210?ef??f(x)dx10101??(x?1)10e?xdx??(x?1)10e?x?5?f10e120?2x010??10(x?1)9e?xdx012?[?(x?1)edx]1234?7(?2)4e5。证明f?g?f?g 证明:
f?(f?g)?g?f?g?g?f?g?f?g6。对f(x),g(x)?C[a,b],定义
1
(1)(f,g)??f?(x)g?(x)dxab(2)(f,g)??f?(x)g?(x)dx?f(a)g(a)ab
问它们是否构成积。 解:
(1)令f(x)?C(C为常数,且C?0)
则f?(x)?0
.. ..
数值分析第五版_李庆扬_王能超_易大义主编课后习题答案
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