15 函数逼近

1111 函数逼近与希尔伯特矩阵函数逼近与希尔伯特矩阵函数逼近与希尔伯特矩阵函数逼近与希尔伯特矩阵 切比雪夫多项式切比雪夫多项式切比雪夫多项式切比雪夫多项式 勒让德多项式勒让德多项式勒让德多项式勒让德多项式 正交多项式的应用正交多项式的应用正交多项式的应用正交多项式的应用 函数逼近 2222 问题问题问题问题. . . . 求二次多项式求二次多项式求二次多项式求二次多项式 PPPP((((xxxx)= )= )= )= aaaa0000 + + + + aaaa1111x + ax + ax + ax + a2222xxxx2222 使使使使 minminminmin)])])])]sin(sin(sin(sin())))(((([[[[ 1111 0000 2222 ====−−−−∫∫∫∫ dxdxdxdxxxxxxxxxPPPP ππππ 0 0.5 1 0 1 连续函数的最佳平方逼近连续函数的最佳平方逼近连续函数的最佳平方逼近连续函数的最佳平方逼近 已知已知已知已知 ffff((((xxxx))))∈CCCC[0, 1], [0, 1], [0, 1], [0, 1], 求多项式求多项式求多项式求多项式 PPPP((((xxxx) = ) = ) = ) = aaaa0000 + a + a + a + a1111x + ax + ax + ax + a2 2 2 2 xxxx2222 + + + + …………………… + + + + aaaannnn x x x x nnnn 使得使得使得使得 minminminmin)])])])](((())))(((([[[[ 1111 0000 2222 ====−−−−==== ∫∫∫∫ dxdxdxdxxxxxffffxxxxPPPPLLLL ∫∫∫∫ ∑∑∑∑ ==== −−−−==== 1111 0000 2222 0000 11110000 ]]]]))))(((([[[[)))),,,,,,,,,,,,(((( dxdxdxdxxxxxffffxxxxaaaaaaaaaaaaaaaaLLLL n nn n j jj j j jj j j jj jn nn n ⋯令令令令 ∑∑∑∑ ∫∫∫∫∫∫∫∫∫∫∫∫ ∑∑∑∑ ======== ++++−−−−==== n nn n j jj j j jj j j jj j n nn n j jj j j jj j j jj j dx dxdx dxx xx xf ff fdx dxdx dxx xx xf ff fx xx xa aa adx dxdx dxx xx xa aa aL LL L 0000 1111 0000 2222 1111 0000 1111 0000 2222 0000 )])])])](((([[[[))))((((2222]]]][[[[ 函数逼近与希尔伯特矩阵函数逼近与希尔伯特矩阵函数逼近与希尔伯特矩阵函数逼近与希尔伯特矩阵 3333 ∫∫∫∫∑∑∑∑ ∫∫∫∫ −−−−====∂∂∂∂ ∂∂∂∂ ==== ++++ 1111 0000 0000 1111 0000 ))))((((22222222 dxdxdxdxxxxxffffxxxxdxdxdxdxxxxxaaaa a aa a L LL L k kk k n nn n j jj j k kk kj jj j j jj j k kk k ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎦⎦⎦⎦ ⎤⎤⎤⎤ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎣⎣⎣⎣ ⎡⎡⎡⎡ ==== ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎦⎦⎦⎦ ⎤⎤⎤⎤ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎣⎣⎣⎣ ⎡⎡⎡⎡ ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎥⎥⎥⎥ ⎦⎦⎦⎦ ⎤⎤⎤⎤ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎣⎣⎣⎣ ⎡⎡⎡⎡ ++++++++ ++++ ++++ n nn nn nn n b bb b b bb b b bb b a aa a a aa a a aa a n nn nn nn n n nn n n nn n ⋮⋮ ⋯⋯ ⋯⋯⋯⋯ ⋯ ⋯ 1111 0000 1111 0000 ))))11112222/(/(/(/(1111))))1111/(/(/(/(1111 ))))2222/(/(/(/(11113333////11112222////1111 ))))1111/(/(/(/(11112222////11111111 系数矩阵被称为系数矩阵被称为系数矩阵被称为系数矩阵被称为HilbertHilbertHilbertHilbert矩阵矩阵矩阵矩阵 0000==== ∂∂∂∂ ∂∂∂∂ k kk k a aa a L LL L令令令令 ∫∫∫∫==== 1111 0000 ))))(((( dxdxdxdxxxxxffffxxxxbbbb kkkk k kk k 记记记记 ∑∑∑∑ ∫∫∫∫∫∫∫∫∫∫∫∫ ∑∑∑∑ ======== ++++−−−−==== n nn n j jj j j jj j j jj j n nn n j jj j j jj j j jj j dx dxdx dxx xx xf ff fdx dxdx dxx xx xf ff fx xx xa aa adx dxdx dxx xx xa aa aL LL L 0000 1111 0000 2222 1111 0000 1111 0000 2222 0000 )])])])](((([[[[))))((((2222]]]][[[[ 4444 定义定义定义定义6.3 6.3 6.3 6.3 设设设设 ffff((((xxxx), ), ), ), gggg((((xxxx))))∈C[C[C[C[a, ba, ba, ba, b], ], ], ], ρ((((xxxx))))是区间是区间是区间是区间[[[[aaaa,,,,bbbb]]]]上的上的上的上的 权函数权函数权函数权函数,,,,若等式若等式若等式若等式 0000))))(((())))(((())))(((()))),,,,(((( ======== ∫∫∫∫ b bb b a aa a dx dxdx dxx xx xg gg gx xx xf ff fx xx xg gg gf ff f ρ ρρ ρ 成立成立成立成立,,,,则称则称则称则称ffff((((xxxx), ), ), ), gggg((((xxxx))))在在在在[[[[a, ba, ba, ba, b]]]]上带权上带权上带权上带权ρ((((xxxx))))正交正交正交正交.... 当当当当ρ((((xxxx)=1)=1)=1)=1时时时时,,,,简称正交简称正交简称正交简称正交。 例例例例1 1 1 1 验证验证验证验证 ϕϕϕϕ0000((((xxxx)=1,)=1,)=1,)=1, ϕϕϕϕ1111((((xxxx)=)=)=)=xxxx 在在在在[ [ [ [ ––––1, 1]1, 1]1, 1]1, 1]上正交上正交上正交上正交,,,, 并求二次多项式并求二次多项式并求二次多项式并求二次多项式 ϕϕϕϕ2222((((xxxx) ) ) ) 使之与使之与使之与使之与ϕϕϕϕ0000((((xxxx), ), ), ), ϕϕϕϕ1111((((xxxx))))正交正交正交正交 00001111))))(((())))(((( 1111 1111 1111 1111 11110000 ====⋅⋅⋅⋅==== ∫∫∫∫∫∫∫∫ −−−−−−−− xdxxdxxdxxdxdxdxdxdxxxxxxxxx ϕϕϕϕϕϕϕϕ解解解解:::: 5555 设设设设 ϕϕϕϕ2222((((xxxx) =) =) =) = x x x x2222 + + + + aaaa21212121xxxx + + + + aaaa22 22 22 22 0000))))((((1111 1111 1111 2222 ====⋅⋅⋅⋅∫∫∫∫−−−− dxdxdxdxxxxxϕϕϕϕ 0000))))(((( 1111 1111 2222 ====∫∫∫∫−−−− dxdxdxdxxxxxxxxxϕϕϕϕ 3333 1111 ))))(((( 22222222 −−−−==== xxxxxxxxϕϕϕϕ所以所以所以所以, , , , 0000))))(((( 1111 1111 22 22222221212121 2222 ====++++++++∫∫∫∫−−−− dxdxdxdxaaaaxxxxaaaaxxxx 0000))))(((( 1111 1111 22 22222221212121 2222 ====++++++++∫∫∫∫−−−− dxdxdxdxaaaaxxxxaaaaxxxxxxxx a aa a22222222= - 1/3= - 1/3= - 1/3= - 1/3 aaaa21212121=0=0=0=0 2/3+22/3+22/3+22/3+2aaaa22222222 = 0 = 0 = 0 = 0 2222aaaa21212121/3=0/3=0/3=0/3=0 6666 切比雪夫多项式切比雪夫多项式切比雪夫多项式切比雪夫多项式 TTTT0000((((xxxx)=1, )=1, )=1, )=1, TTTT1111((((xxxx)= cos)= cos)= cos)= cosθθθθ = = = = xxxx, , , , TTTT2222((((xxxx)=cos2)=cos2)=cos2)=cos2θθθθ ························ T TT T n nn n ((((xxxx)=cos()=cos()=cos()=cos(nnnnθθθθ),),),),···································· 有有有有 cos(n+1) cos(n+1) cos(n+1) cos(n+1)θθθθ=2 cos=2 cos=2 cos=2 cosθθθθ cos(n cos(n cos(n cos(nθθθθ)))) –––– cos(n-1) cos(n-1) cos(n-1) cos(n-1)θθθθ ，从而 T TT T n+ n+n+ n+1111((((xxxx) = 2) = 2) = 2) = 2 x T x T x T x Tnnnn((((xxxx) ) ) ) –––– TTTTnnnn-1-1-1-1((((xxxx)))) ((((nnnn ≥ 1) 1) 1) 1) 所以所以所以所以, , , , TTTT0000((((xxxx)=1, )=1, )=1, )=1, TTTT1111((((xxxx)=)=)=)=xxxx, , , , TTTT2222((((xxxx)=2)=2)=2)=2xxxx2222 –––– 1 , 1 , 1 , 1 , ············,,,, TTTT n nn n ((((xxxx)=cos()=cos()=cos()=cos(nnnnarccos(arccos(arccos(arccos(xxxx)))))))),,,,···································· 1.1.1.1.递推公式递推公式递推公式递推公式::::由 cos(n+1) cos(n+1) cos(n+1) cos(n+1)θθθθ++++ cos(n-1) cos(n-1) cos(n-1) cos(n-1)θθθθ =2 cos =2 cos =2 cos =2 cosθθθθ cos(n cos(n cos(n cos(nθθθθ)))) 7777 0000))))cos(cos(cos(cos())))cos(cos(cos(cos( 0000 ====∫∫∫∫ π ππ π θ θθ θθ θθ θθ θθ θ d dd dn nn nm mm m ((((m m m m ≠ n n n n)))) 0000coscoscoscoscoscoscoscos ))))(((())))(((( 1111 1111 )))),,,,(((( 0000 1111 1111 2222 ======== −−−− ==== ∫∫∫∫ ∫∫∫∫−−−− π ππ π θ θθ θθ θθ θθ θθ θ d dd dn nn nm mm m dx dxdx dxx xx xT TT Tx xx xT TT T x xx x T TT TT TT T n nn nm mm mn nn nm mm m 所以所以所以所以,,,,切比雪夫多项式在切比雪夫多项式在切比雪夫多项式在切比雪夫多项式在[[[[–––– 1 1 1 1 , , , , 1]1]1]1]上带权上带权上带权上带权 正交正交正交正交 22221111 1111 ))))(((( x xx x x xx x −−−− ====ρρρρ 2.2.2.2.切比雪夫多项式的正交性切比雪夫多项式的正交性切比雪夫多项式的正交性切比雪夫多项式的正交性 8888 3.3.3.3.切比雪夫多项式零点切比雪夫多项式零点切比雪夫多项式零点切比雪夫多项式零点 n nn n 阶阶阶阶ChebyshevChebyshevChebyshevChebyshev多项式多项式多项式多项式: : : : TTTT n nn n =cos(=cos(=cos(=cos(nnnnθθθθ), ), ), ), 或或或或, , , , TTTT n nn n (((( x x x x ) = cos() = cos() = cos() = cos(n n n n arccosarccosarccosarccos x x x x )))) 2222 ))))11112222(((( arccosarccosarccosarccos π ππ π++++ ==== k kk k x xx xn nn n ((((kkkk=0,1,=0,1,=0,1,=0,1,············,n-1 ),n-1 ),n-1 ),n-1 )取取取取 T TT T1111=cos=cos=cos=cosθθθθ====xxxx )))) 2222 ))))11112222(((( cos(cos(cos(cos( n nn n k kk k x xx x k kk k π ππ π++++ ====即即即即 ((((kkkk=0,1,=0,1,=0,1,=0,1,············,n-1 ),n-1 ),n-1 ),n-1 ) 9999 4.4.4.4.切比雪夫多项式的极性切比雪夫多项式的极性切比雪夫多项式的极性切比雪夫多项式的极性 T TT T n nn n ((((xxxx) ) ) ) 的最高次项的最高次项的最高次项的最高次项 xxxxnnnn 的系数为的系数为的系数为的系数为 2 2 2 2n n n n –––– 1 1 1 1 所有最高次项系数为所有最高次项系数为所有最高次项系数为所有最高次项系数为1111的的的的nnnn次多项式中次多项式中次多项式中次多项式中,,,, P P P P n nn n ((((xxxx)= 2)= 2)= 2)= 21 1 1 1 –––– n n n n TTTT n nn n ((((xxxx)))) 则则则则 minminminmin||||))))((((||||maxmaxmaxmax 11111111 ==== ≤≤≤≤≤≤≤≤−−−− x xx xP PP P n nn n x xx x 例如例如例如例如 tttt k kk k = = = = – –– –1+0.21+0.21+0.21+0.2k k k k (((( k = k = k = k = 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, ············, 10), 10), 10), 10) )))) 22222222 ))))11112222(((( cos(cos(cos(cos( π ππ π++++ ==== k kk k x xx x k kk k (((( k = k = k = k = 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, ············, 10), 10), 10), 10) ���� P P P P11111111((((xxxx)=()=()=()=(x x x x –––– x x x x0000)()()()(x x x x –––– x x x x1111))))························((((x x x x –––– x x x x10101010)))) QQQQ11111111((((xxxx)=()=()=()=(x x x x –––– tttt0000)()()()(x x x x –––– tttt1111))))························((((x x x x –––– tttt10101010)))) 10101010 勒让德勒让德勒让德勒让德(Legendre)(Legendre)(Legendre)(Legendre)多项式多项式多项式多项式 1.1.1.1. 表 关于同志近三年现实表现材料材料类招标技术评分表图表与交易pdf视力表打印pdf用图表说话 pdf 达式表达式表达式表达式 PPPP0000((((xxxx) = 1, ) = 1, ) = 1, ) = 1, PPPP1111((((xxxx) = ) = ) = ) = xxxx ]]]]))))1111[([([([( !!!!2222 1111 ))))(((( 2222 nnnn n nn n n nn n n nn n n nn n x xx x dx dxdx dx d dd d n nn n x xx xP PP P −−−−==== ((((nnnn ≥ 1) 1) 1) 1) 2. 2. 2. 2. 正交性正交性正交性正交性 ⎪⎪⎪⎪⎩⎩⎩⎩ ⎪⎪⎪⎪ ⎨⎨⎨⎨ ⎧⎧⎧⎧ ==== ++++ ≠≠≠≠ ====∫∫∫∫−−−− n nn nm mm m n nn n n nn nm mm m dx dxdx dxx xx xP PP Px xx xP PP P n nn nm mm m ,,,, 11112222 2222 ,,,,0000 ))))(((())))(((( 1111 1111 11111111 3.3.3.3.递推式递推式递推式递推式 ⎪⎩ ⎪ ⎨ ⎧ + − + + = == −+ 11111111 11110000 11111111 11112222 1111 n nn nn nn nn nn n p pp p n nn n n nn n xp xpxp xp n nn n n nn n p pp p x xx xp pp pp pp p ,, )()( 11113333 2222 1111 2222 2222 −= xxxxxxxxpppp )()( xxxxxxxxxxxxpppp 333355552222 1111 3333 3333 −= 4.4.4.4.零点分布零点分布零点分布零点分布 P PP P n nn n ((((xxxx) ) ) ) 的的的的nnnn 个零点个零点个零点个零点,,,,落入区间落入区间落入区间落入区间[ [ [ [ ––––1, 1]1, 1]1, 1]1, 1]中中中中 P PP P2222((((xxxx))))的两个零点的两个零点的两个零点的两个零点:::: P PP P3333((((xxxx))))的三个零点的三个零点的三个零点的三个零点:::: 3333 1111 1111 −−−−====xxxx 3333 1111 2222 ====xxxx 5555 3333 1111 −−−−====xxxx 00002222 ====xxxx 5555 3333 3333 ====xxxx 12121212 用正交多项式作最佳平方逼近用正交多项式作最佳平方逼近用正交多项式作最佳平方逼近用正交多项式作最佳平方逼近 设设设设 P PP P0000((((xxxx), ), ), ), PPPP1111((((xxxx), ), ), ), ············,,,,PPPPnnnn((((xxxx))))为区间为区间为区间为区间[[[[aaaa , , , , bbbb]]]]上的正交上的正交上的正交上的正交 多项式多项式多项式多项式, , , , 即即即即 0000))))(((())))(((()))),,,,(((( ======== ∫∫∫∫ b bb b a aa a j jj jk kk kj jj jk kk k dx dxdx dxx xx xP PP Px xx xP PP PP PP PP PP P ((((k k k k ≠ j j j j , , , , kkkk, , , , j =j =j =j = 0,1,0,1,0,1,0,1,············, , , , n n n n )))) 求求求求 PPPP((((xxxx) = ) = ) = ) = aaaa0000PPPP0000((((xxxx) + ) + ) + ) + aaaa1111PPPP1111((((xxxx) + ) + ) + ) + ············ + + + + aaaannnnPPPPnnnn((((xxxx)))) minminminmin)])])])](((())))(((([[[[ 2222 ====−−−−==== ∫∫∫∫ b bb b a aa a dx dxdx dxx xx xf ff fx xx xP PP PL LL L 使使使使 ∫∫∫∫ ∑∑∑∑ −−−−==== ==== b bb b a aa a n nn n j jj j j jj jj jj jn nn n dx dxdx dxx xx xf ff fx xx xP PP Pa aa aa aa aa aa aa aa aL LL L 2222 0000 11110000 )])])])](((())))(((([[[[)))),,,,,,,,,,,,(((( ⋯ 13131313 )))),,,,(((( )))),,,,(((( k kk kk kk k k kk k k kk k P PP PP PP P f ff fP PP P a aa a ==== ((((k = k = k = k = 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, ············, , , , n n n n )))) dx dxdx dxx xx xf ff fx xx xP PP Pa aa ax xx xP PP P a aa a L LL L n nn n j jj j j jj jj jj j b bb b a aa a k kk k k kk k ∑∑∑∑∫∫∫∫ ==== −−−−==== ∂∂∂∂ ∂∂∂∂ 0000 )])])])](((())))(((([[[[))))((((2222 0000==== ∂∂∂∂ ∂∂∂∂ k kk k a aa a L LL L令令令令 ∫∫∫∫ b bb b a aa a k kk k dx dxdx dxx xx xf ff fx xx xP PP P ))))(((())))((((记记记记 ( ( ( (PPPP k k k k , , , , ffff ) = ) = ) = ) = ))))((((,,,,0000))))(((())))(((()))),,,,(((( jjjjkkkkdxdxdxdxxxxxPPPPxxxxPPPPPPPPPPPP b bb b a aa a j jj jk kk kj jj jk kk k ≠≠≠≠======== ∫∫∫∫由于由于由于由于 则有则有则有则有 )))),,,,(((()))),,,,(((( ffffPPPPaaaaPPPPPPPP k kk kk kk kk kk kk kk k ==== ((((k = k = k = k = 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, ············, , , , n n n n )))) ∑∑∑∑ ==== ==== n nn n k kk k k kk k k kk kk kk k k kk k x xx xP PP P P PP PP PP P f ff fP PP P x xx xP PP P 0000 ))))(((( )))),,,,(((( )))),,,,(((( ))))(((( ffff((((xxxx))))的平方逼近的平方逼近的平方逼近的平方逼近 14141414 构造区间构造区间构造区间构造区间[0[0[0[0，1]1]1]1]上的正交多项式上的正交多项式上的正交多项式上的正交多项式 P P P P0000((((xxxx)= 1)= 1)= 1)= 1，PPPP1111((((xxxx)=)=)=)= x x x x –––– 1/21/21/21/2，PPPP2222((((xxxx)= )= )= )= xxxx2222 –––– xxxx + 1/6 + 1/6 + 1/6 + 1/6 例例例例 求二次多项式求二次多项式求二次多项式求二次多项式 PPPP((((xxxx)= )= )= )= aaaa0000 + + + + aaaa1111x + ax + ax + ax + a2222xxxx2222 使使使使 minminminmin)])])])]sin(sin(sin(sin())))(((([[[[ 1111 0000 2222 ====−−−−∫∫∫∫ dxdxdxdxxxxxxxxxPPPP ππππ ))))(((( )))),,,,(((( ))))))))sin(sin(sin(sin(,,,,(((( ))))(((( )))),,,,(((( ))))))))sin(sin(sin(sin(,,,,(((( )))),,,,(((( ))))))))sin(sin(sin(sin(,,,,(((( ))))sin(sin(sin(sin( 2222 22222222 2222 1111 11111111 1111 00000000 0000 x xx xP PP P P PP PP PP P x xx xP PP P x xx xP PP P P PP PP PP P x xx xP PP P P PP PP PP P x xx xP PP P x xx x π ππ ππ ππ π π ππ π π ππ π ++++++++≈≈≈≈ 1111 ////2222 )))),,,,(((( ))))))))sin(sin(sin(sin(,,,,(((( 00000000 0000 ππππππππ ==== P PP PP PP P x xx xP PP P 12121212////1111 0000 )))),,,,(((( ))))))))sin(sin(sin(sin(....(((( 11111111 1111 ==== P PP PP PP P x xx xP PP P π ππ π 180180180180////1111 3333////))))12121212(((( )))),,,,(((( ))))))))sin(sin(sin(sin(....(((( 33332222 22222222 2222 ππππππππππππ −−−−==== P PP PP PP P x xx xP PP P ⎪⎩ ⎪ ⎨ ⎧ + − + + = == −+ 11111111 11110000 11111111 11112222 1111 n nn nn nn nn nn n p pp p n nn n n nn n xp xpxp xp n nn n n nn n p pp p x xx xp pp pp pp p ,, 15151515 最佳平方逼近最佳平方逼近最佳平方逼近最佳平方逼近:::: )))) 6666 1111 ((((1225122512251225....4444 2222 ))))sin(sin(sin(sin( 2222 ++++−−−−−−−−≈≈≈≈ xxxxxxxxxxxx π ππ π π ππ π 0 0.5 1 0 1 )))) 6666 1111 ((((1225122512251225....4444 2222 ))))(((( 2222 ++++−−−−−−−−==== xxxxxxxxxxxxPPPP π ππ π ))))sin(sin(sin(sin())))(((( xxxxxxxxffff ππππ====