[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 : z) C6 g7 [6 y1 {) e9 L7 c in the sequence converges. Assume further that there exists a function g such that . |/ c. z: P) g% ?8 ^ uniformly on & k* C, [7 Q" l! c3 X . Then:

a) There exists a function f such that , L% I/ s# l1 e8 q uniformly on r1 \0 n6 h O' \" {6 r .

b) For each x in K% p0 Z2 Y; C) ?% ~ the derivative exists and equal .

Proof. Assume that 9 E; p0 D9 O- `0 _) [; q and define a new sequence as follows:

(8)

The sequence . q0 q2 O$ U, N so formed depends on the choice of c. Convergence of follows from the hypothesis, since . We will prove next that converges uniformly on * c: C N% N; t& i. { . If , we have

6 d% E3 e; m; B8 c8 P" `

, 3 D; W3 p( L7 D, [2 C4 R (9)

where 2 a) j4 V7 _7 [1 X; [ . Now + {! k4 t0 Q0 c; _) t7 l# K% } exists for each x in 7 t* J6 r; u" X' Z. F and has the value . Applying the Mean-Value Theorem in (9), we get

, , {1 g0 Q! }1 U0 H (10)

where / f1 o. d1 l, S' D- q) N lies between x and c. Since ; E( S) y" a+ R2 O" i: ` converges uniformly on 7 z1 r( F% i( Q" M (by hypothesis), we can use (10), together with the Cauthy condition, to deduce that converges uniformly on , V$ y8 H+ W* U" K4 E2 W .

Now we can show that converges uniformly on . Let us form the particular sequence R& ^- h& J8 W. ?. r8 o1 d corresponding to the special point # h& \. Q! g9 p( M3 s for which # Q# ^6 b3 A5 z( f" ~0 d+ c. C6 L is assumed to converge. Form (8) we can write

an equation which holds for every x in . Hence we have

This equation, with the help of the Cauthy condition, establishes the uniform convergence of on # m0 `, p2 Z' b( i+ J. w7 B4 E6 G" G . This proves (a).

To prove (b), return to the sequence 7 J% R5 f& Z4 Z' A& ? defined by (8) for an arbitrary point c in 8 o X, e, y: f1 R5 i- Z! { and let 8 \5 P% U' S. d) E+ N; m: t . The hypothesis that 2 `% p9 H2 }) A- M( [5 h2 m2 A exists means that . In other words, each is continuous at c. Since uniformly on , 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 ! B! D9 @ W" ]1 ^2 N exists and equals . But

hence . Since c is an arbitrary point of 1 P; \8 u* A* c3 j' L- j , this proves (b).

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

# N- M% p9 B# |3 U1 l

Theorem 9.14. Assume that each is a real-valued function defined on such that the derivative ! \0 x9 _4 M H4 y/ M7 d. D$ M exists for each x in & A* L y4 O- h& r+ f7 q . Assume that, for at least one point " _. t/ g5 |: g: n in , the series converges. Assume further that there exists a function g such that (uniformly on ). Then:

a) 2 y2 @/ o3 |2 O; C There exists a function f such that (uniformly on 8 n/ y. e3 u. d* [, o ).

b) If , the derivative exists and equals .

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？
/ n( M$ T: l! Y" t8 ^

[此贴子已经被作者于2008-7-28 14:24:38编辑过]

rao

看不清楚。。。。。