[功課] 如果你沒時間看Bernstein

[jacky7987 (憶)]

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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※ 編輯: jacky7987       來自: 123.193.93.138       (01/11 01:13)
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