[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 6 X$ ^5 H9 T9 H6 s* C in & C" O- v2 G3 S the sequence converges. Assume further that there exists a function g such that # ?, Q' T4 k) }0 S5 v: S6 K0 [' B uniformly on . Then:

a) There exists a function f such that uniformly on ( N4 ?* ~$ S1 I9 j4 ?. l, F .

b) For each x in B3 m& z" I* A the derivative " k# Z7 P$ r+ k# X1 Z6 r' v exists and equal .

Proof. Assume that ) b* ^, R/ ` w; W" [/ x5 `6 y2 ~ and define a new sequence as follows:

: {* L- W1 ~* p& k; k (8)

The sequence 8 X" C# l& } j( R9 s4 c so formed depends on the choice of c. Convergence of follows from the hypothesis, since 0 C) r' b) [7 A D( z! m% ~, y! u4 s . We will prove next that converges uniformly on ) I. ] x, U9 Q# ` . If , we have

* [, Z( g0 R7 ?3 V6 d

, (9)

7 `, x- S; w r: U$ q

where . Now ( N) y9 D7 T1 j9 k! ~+ v3 s3 F exists for each x in 9 ?5 M- r4 K% d+ f. I K* A" f) d$ ^ and has the value 1 Q |2 S3 m! N( m1 u2 @ . Applying the Mean-Value Theorem in (9), we get

, . ^7 e6 A c9 V5 Z1 r+ X (10)

where + a/ J2 B! U5 b( x4 S% q' P lies between x and c. Since converges uniformly on * }. {1 b% T; Y# f1 Q b+ } (by hypothesis), we can use (10), together with the Cauthy condition, to deduce that 9 A& s. w4 S3 C converges uniformly on .

Now we can show that converges uniformly on / g4 t# H, i3 }: j; F . Let us form the particular sequence 1 ]: y1 A$ c y corresponding to the special point 9 n5 i5 N, k6 r for which 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 . This proves (a).

To prove (b), return to the sequence 0 j" g- p" \4 f+ S1 J$ z4 h( B defined by (8) for an arbitrary point c in and let . The hypothesis that exists means that . In other words, each is continuous at c. Since $ p k' s8 }2 V! x, ?: k5 V+ B uniformly on , the limit function G is also continuous at c. This means that

7 G5 O. j+ W0 L+ {! A% k (11)

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

Hence, (11) states that the derivative exists and equals / P! h! D7 n5 k3 Q( w$ [2 ?) Y . But

hence ! j6 U2 `; X8 {) o% O7 D . Since c is an arbitrary point of 9 b% i- X' O# d( R K$ Q , this proves (b).

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

Theorem 9.14. Assume that each is a real-valued function defined on # i$ F6 r; `& ~3 c such that the derivative 3 ~6 n. @, N+ G q exists for each x in . Assume that, for at least one point in , the series ) Z2 _8 }& T. o1 O" I1 R( d7 M' { converges. Assume further that there exists a function g such that (uniformly on 0 R- v1 k8 Y; P. h ). Then:

a) There exists a function f such that + [: q) u U v (uniformly on ! |; ~! i0 Y. X, D' E. W+ N ).

b) 4 C$ N) D6 `2 c" I; W! B If , the derivative exists and equals - M2 M* e+ R, _) Q( d( X .

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。。。。。