[求助]谁能帮我翻译一下这篇文章?~

angel_lys

文章是讲一致收敛和微分的~我英语不好看不明白~请大家帮帮忙啊~~~~

Theorem 9.13. Assume that each term of is a real-valued function having a finite derivative at each point of an open interval . Assume that for at least one point ( w8 s) H2 e2 o) I9 t, S8 B in the sequence converges. Assume further that there exists a function g such that uniformly on ; c3 L% T- l+ u. I+ ]' ~ . Then:

4 ]1 K" L5 v( o2 K2 w7 }; |$ i

a) There exists a function f such that B% u! a. {2 q uniformly on .

b) For each x in " ~9 q+ c$ A+ t9 d: ?* m the derivative " O0 a$ W) c6 U exists and equal .

Proof. Assume that 7 c* D* W2 |* h y and define a new sequence as follows:

! F+ K! y! @. M7 C% M

6 }- @# s' I9 m9 x2 @ (8)

; n0 T% _: ]( @9 K. @! k

The sequence so formed depends on the choice of c. Convergence of follows from the hypothesis, since 6 `& F/ t: F0 K . We will prove next that # \( r& W5 v0 d0 [ converges uniformly on . If , we have

, (9)

where . Now & b- @: s5 U! S4 | S, M5 U; C& x exists for each x in and has the value 7 M4 o) F$ l* a$ N' M . Applying the Mean-Value Theorem in (9), we get

, " @9 l& h, ~' R/ b) r - o# q' f& [; \0 U (10)

where 4 w# G. r3 b4 l9 } lies between x and c. Since / k7 B1 ]7 B+ n' } converges uniformly on ( v. u& m* U' j4 r; q (by hypothesis), we can use (10), together with the Cauthy condition, to deduce that converges uniformly on 8 q5 s7 w6 T+ w" k, ]' F# `* s .

Now we can show that $ P! G6 f; U& B+ ~7 X converges uniformly on . Let us form the particular sequence 1 v- J8 o9 u8 ~$ x5 X corresponding to the special point for which P z7 U! n) e' h/ r4 s9 s is assumed to converge. Form (8) we can write

an equation which holds for every x in 4 v/ [) q! ^9 F5 f& r" O . Hence we have

This equation, with the help of the Cauthy condition, establishes the uniform convergence of on . This proves (a).

To prove (b), return to the sequence $ Z% @* M; z! v/ f defined by (8) for an arbitrary point c in 5 q- ]; ^6 e3 Y and let . The hypothesis that exists means that . In other words, each is continuous at c. Since uniformly on 8 w' G) ]! H# `) l , the limit function G is also continuous at c. This means that

(11)

the existence of the limit being part of the conclusion. But for , we have

Hence, (11) states that the derivative exists and equals . But

hence . Since c is an arbitrary point of 2 F. ]3 G4 M$ A& L8 D , this proves (b).

When we reformulate Theorem 9.13 in terms of series, we obtain

Theorem 9.14. Assume that each 8 O' G- |8 m% a2 _ v4 f is a real-valued function defined on such that the derivative exists for each x in 1 u. J$ B' H4 B# o . Assume that, for at least one point in , the series converges. Assume further that there exists a function g such that (uniformly on ). Then:

a) There exists a function f such that & R7 \8 ~( f5 t# ]% b) k5 C (uniformly on - J+ O" c% c. Y ).

b) If , the derivative exists and equals 2 \' D3 z; G# D0 Q/ b' w4 A .

lzh0601

我怎么只看到脚本文件呢？

tinkertinker

 “Theorem 9.13. Assume that each term of is a。。。。。” 这里面有没有漏了字？

[此贴子已经被作者于2008-7-28 14:24:38编辑过]

rao

看不清楚。。。。。