Bernstein 多项式 Bernstein 逼近定理 Weierestrass 逼近定理 保形性质 Bezier 曲线 §7.5用多项式一致逼近连续函数 设F是定义在区间I上的一个函数空间.对于定义在I上的一个函数 f,如果对任意正数ε都存在g∈F使得 |f(x)-g(x)|<ε，(x∈I) 则称可以用F中的函数一致逼近f.此时，存在F中的函数列{gn}在I上 一致收敛于 f. 由于多项式具有良好的性质，我们希望可以用多项式来一致逼近函数. 注意到多项式是连续的，可以用多项式来一致逼近的函数一定也是连续的. 我们要讨论下面的问题： 问题对于有限闭区间[a,b]上的连续函数f(x),是否存在多项式函数 列{fn(x)}在[a,b]上一致收敛于f(x)? 11 返回 全屏 关闭 退出 1/13
Bernstein õª Bernstein %C½n Weierestrass %C½n /5 Bezier §7.5 ^õª%CëY¼ê F ´½Â3«m I þ¼êm. éu½Â3 I þ¼ê f, XJé?¿ê ε Ñ3 g ∈ F ¦ |f(x) − g(x)| < ε, (∀ x ∈ I) K¡±^ F ¥¼ê%C f. d, 3 F ¥¼ê {gn} 3 I þ Âñu f. duõªäkûÐ5, ·F"±^õª5%C¼ê. 5¿õª´ëY, ±^õª5%C¼ê½´ëY. ·?Øe¡¯K: ¯K éuk4«m [a, b] þëY¼ê f(x), ´Ä3õª¼ê {fn(x)} 3 [a, b] þÂñu f(x)? 1/13 kJ Ik J I £ ¶ '4 òÑ
Bernstein õª Bernstein %C½n Weierestrass %C½n /5 Bezier Bn i (x) = n k x k (1 − x) n−k (i = 0, 1, · · · , n) ¡ Bernstein ļê. ½Â 1 f(x) ´ [0, 1] þëY¼ê, ¡ Bn(f; x) = X n i=0 f( i n )Bn i (x), (1) f n Bernstein õª (1912 cJÑ). w, Bn(f, x) ´gê 6 n õª, Bn(1; x) = 1. (2) ±y² Bn(x; x) = x. (3) 2/13 kJ Ik J I £ ¶ '4 òÑ
Bernstein õª Bernstein %C½n Weierestrass %C½n /5 Bezier Bn(x; x) = X n i=0 i n n i x i (1 − x) n−i = X n i=1 (n − 1)! (i − 1)!(n − i)! x i (1 − x) n−i = X n−1 i=0 (n − 1)! i!(n − 1 − i)! x i+1(1 − x) n−1−i = x X n−1 i=0 n − 1 i x i (1 − x) n−1−i = x. ^aq{±y² Bn(x 2 ; x) = (1 − 1 n )x 2 + 1 n x. (4) ±y, f ´ m gõª, Bn(f; x) ´gê min(n, m) õ ª. d (2), (3), (4) ± X n k=0 (k − nx) 2Bn k (x) = nx(1 − x). (5) 3/13 kJ Ik J I £ ¶ '4 òÑ
Bernstein õª Bernstein %C½n Weierestrass %C½n /5 Bezier ½n 1 (Bernstein) f(x) ´ [0, 1] þëY¼ê, K Bn(f; x) 3 [0, 1] þ Âñu f(x). y² éuê δ, ω(δ) f ëY , =, ω(δ) = max |x−y|6δ x,y∈[0,1] |f(x) − f(y)|. ½ÂKê: λ(x, y; δ) = |x − y| δ (êÜ©). Kk |f(x) − f(y)| 6 1 + λ(x, y; δ) ω(δ). Ï f(x) = X n k=0 f(x)Bn i (x), Bn(f; x) = X n k=0 f( k n )Bn i (x), 4/13 kJ Ik J I £ ¶ '4 òÑ
Bernstein õª Bernstein %C½n Weierestrass %C½n /5 Bezier ¤± |Bn(f; x) − f(x)| = X n k=0 f( k n ) − f(x) Bn i (x) 6 X n k=0 f( k n ) − f(x) Bn i (x) 6 X n k=0 1 + λ( k n , x; δ) ω(δ)Bn i (x) 6 ω(δ) 1 +X n k=0 λ 2 · Bn i (x) ! 6 ω(δ) 1 +X n k=0 (x − k n ) 2 δ 2 Bn i (x) ! = ω(δ) 1 + x(1 − x) nδ2 6 ω(δ) 1 + 1 4nδ2 . 5/13 kJ Ik J I £ ¶ '4 òÑ
Bernstein õª Bernstein %C½n Weierestrass %C½n /5 Bezier δ = √ 1 n , =, |Bn(f; x) − f(x)| 6 5 4 ω 1 √ n . (6) ¤±k lim n→∞ Bn(f; x) = f(x) 3 [0, 1] þ¤á, =, ùÒ`² Bn(f; x) 3 [0, 1] þÂñu f(x). y . 5 (6) ªØ=y²ëY¼ê Bernstein õªÂñùëY¼ ê, Ñ ÂñÝ. §L²ùÝØL ω(n −1/2 ). ùÝ´ éú. 6/13 kJ Ik J I £ ¶ '4 òÑ
Bernstein õª Bernstein %C½n Weierestrass %C½n /5 Bezier f(x) ´k4«m [a, b] þëY¼ê, - g(x) = f(a + x(b − a)), x ∈ [0, 1]. K g(x) 3 [0, 1] ëY, Ïd Bn(g; x) 3 [0, 1] þÂñu g(x), l y² Bn(g; x−a b−a ) 3 [a, b] þÂñu f(x). =, k ½n 2 (Weierestrass) k4«mþëY¼ê^õª%C. Bernstein ½n´ Weierestrass ½nE5y², Bernstein õª ÂñuëY¼êÝ'ú, ^5ëY¼êCqØÜ·. ù ´o3éãmS, Bernstein õªvkoA^Ï. ØL Bernstein õªkéÐ/5. 7/13 kJ Ik J I £ ¶ '4 òÑ
Bernstein õª Bernstein %C½n Weierestrass %C½n /5 Bezier ½n 3 (Bernstein õª/5) f(x) ´½Â3 [0, 1] þ¼ê. Kk 1 ◦ Bn(f; 0) = f(0), Bn(f; 1) = f(1); (à:) 2 ◦ e f 3 [0, 1] þK, K Bn(f; x) 3 [0, 1] þK; (Ò) 3 ◦ e f 3 [0, 1] þüN4O, K Bn(f; x) 3 [0, 1] þüN4O;(üN) 4 ◦ e f ´ [0, 1] þà¼ê, K Bn(f; x) ´ [0, 1] þà¼ê. (à) «m [0, 1] ©:: i n , (i = 0, 1, · · · , n) ¡ Bernstein õª!:. !:þ fi = f( i n ) ¡!:. ¿P ∆f i n = f i + 1 n − f i n , (0 6 i 6 n − 1) ∆2f i n = f i + 2 n − 2f i + 1 n + f i n , (0 6 i 6 n − 2). ©O¡!: ©Ú ©. 8/13 kJ Ik J I £ ¶ '4 òÑ
Bernstein õª Bernstein %C½n Weierestrass %C½n /5 Bezier ·k (Bn(f; x))0 = n X n−1 i=0 ∆f i n B n−1 i (x), (Bn(f; x))00 = n(n − 1)X n−2 i=0 ∆2f i n B n−2 i (x) dd±y²Xe½n: ½n 4 e f ∈ C1 [0, 1], K (Bn(f; x))0 3 [0, 1] þÂñu f 0 (x). e f ∈ C2 [0, 1], K (Bn(f; x))00 3 [0, 1] þÂñu f 00(x). f ´ [0, 1] þ¼ê. P f ∗ i = i n + 1 fi−1 + 1 − i n + 1 fi, (i = 0, 1, · · · , n + 1), ùp f−1 = fn+1 = 0. Kk ½n 5 (, úª) Bn(f; x) = Bn+1(f ∗ ; x). 9/13 kJ Ik J I £ ¶ '4 òÑ
Bernstein õª Bernstein %C½n Weierestrass %C½n /5 Bezier ½n 6 f(x) ´ [0, 1] þà¼ê, @ok Bn(f; x) > Bn+1(f; , x), x ∈ [0, 1], (n = 1, 2, · · ·). ½n 7 (Ziegler, 1968) f(x) ´ [0, 1] þëY¼ê. e Bn(f; x) > Bn+1(f; , x), x ∈ [0, 1], (n = 1, 2, · · ·), K f(x) ´ [0, 1] þà¼ê. ½n 8 (Kelisky-Rivlin ½n) f(x) ´ [0, 1] þëY¼ê. éufS (Bn) (m) := Bn(B(m−1) n ), k lim m→+∞ (Bn) (m) (f; x) = f(0) + (f(1) − f(0))x. d½n¡ Bernstein f 15. 10/13 kJ Ik J I £ ¶ '4 òÑ