看板 Gossiping作者 radiodept ( )標題 Re: [問卦] 我家浴室天花板出現一個洞時間 Tue Jul 2 01:14:38 2013
在下鍵盤湯川教授,人稱台灣伽利略,
讓我用物理來幫你算一下
Consider [-m,m] , m€Natural numbers
By W.A.T., take ε=1/m , there exists Pm s.t. │Pm(x)-f(x)│<1/m for x€[-m,m]
(W.A.T.:if f€C[a,b]
then for all ε>0, there exists polynomial P_ε
s.t. │P_ε(x)-f(x)│<ε for x€[a,b])
Let T_n(x)=:Pn(x)
Then for each x€R, x€[-m,m] for all m≧m* (m* is dependent on x)
hence │Pn(x)-f(x)│< 1/n , for all n≧m* ---(●)
take limit to n, Pn(x)→f(x) #
Let f is a continuous function on R
By W.A.T,
consider
[-1,1], there is a sequence of poly. P s.t. for each ε>0
1,m
|P (x)-f(x)|<ε for x£[-1,1] whenever m≧N for some N
1,m 1 1
˙
˙
˙
(*) [-n,n], there is a sequence of poly. P s.t. for each ε>0
n,m
|P (x)-f(x)|<ε for x£[-n,n] whenever m≧N for some N
n,m n n
˙
˙
˙
Take P = P
n n,Nn
Claim: Pn→f pointwise on R
 ̄ ̄ ̄ ̄ ̄ ̄ ̄ ̄ ̄ ̄ ̄ ̄ ̄ ̄
Given x£R and any ε>0
there is an integer n s.t. x£[-n,n]
then we have |Pn(x)-f(x)| = |P (x)-f(x)|<ε (By (*))
n,Nn
Hence Pn→f pointwise
#
結論:
風壓把你家維修間的天花板掀起來了。
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※ 同主題文章:
Re: [問卦] 我家浴室天花板出現一個洞
07-02 01:14 radiodept.