幫大家簡單整理一下 n r n r n-r 1.B_n(f:x)=sum f(---)( )x (1-x) r=0 n r n r n-r Sometimes we write p (x)=( )x (1-x) r,n r 2.B_n is a linear operator from F([0,1]) to F([0,1]) 翻譯成中文就是把一個定義在[0,1]的函數f 打到 B_n(f)這樣 而且他是線性的i.e. For a in |R B_n(f+a*g:x)=B_n(f:x)+a*B_(g:x) 3.B_n is a monotone(positve) operator i.e. If f(x)≧g(x) in [0,1] => B_n(f:x)≧B_n(g:x) in [0,1] (保持大小的關係) In par, f(x)≧0 => B_n(f:x)≧0 4.(1)The Berstein Polynomial only reproduce linear polynomial i.e. B_n(ax+b:x)=ax+b 就是線性的轉出來會長的一樣 2 2 1 e.g. B_n(x :x )=x +---x(1-x) 沒有出來還是長得一樣 n (2)B_n(f:0)=f(0) B_n(f:1)=f(1) hence Bernstein Polynomial interpolate f at both end of f on [0,1] 5.Defn: The Forward difference operator △ with step size h is defined by ￣￣ △f(x_0)=f(x_0+h)-f(x_0) Under this notation ,the Bernstein Polynomial may be expressed in the form n n r r B_n(f:x)=sum ( )△ f(0)x 1 r=0 r with step size h=--- n 反正就是它可以展成差分的模樣 6.The relation between difference and derivative is m △f(x_0) (m) --------- = f (c) for some c in (x_0,x_0+mh) m h 差分透過補上寬度就可以用均值定理一直簡化到某個c在你差分的範圍裏面 7.Differential form for Bernstein Polynomial (k) (n+k)! n k r n r n-r B (f:x)=-------sum △f(-----)( )x (1-x) 1 n+k n! r=0 n+k r with step size h=----- n+k 反正degree of n+k微分k次就會變成degree of n只是差分距離要用n+k次的而已 8.Suppose f \in C[0,1] ,then for any ε>0 there exists a N\in |N s.t forall n≧N |f(x)-B_n(f:x)|<ε 請記住f一定要定義在[0,1]上而且要連續這個定理才會對 ===== 看到一個機掰的定理不在這個範圍但是很有趣 Voronovskaya’s theorem 1 lim n(f(x)-B_n(f:x))=---x(1-x)f''(x) n->∞ 2 ===== 祝大家期末考順利 --

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