[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 & O' R8 l7 M `; U6 S/ F . Assume that for at least one point in 1 }+ r) x0 D! T2 h3 x W- t the sequence converges. Assume further that there exists a function g such that uniformly on , V3 f* [6 c/ m* a5 ]1 | . Then:

a) There exists a function f such that ' o1 d; Z. t' F uniformly on .

b) For each x in the derivative ' c9 p, \1 v/ D/ ]: g5 r6 @ exists and equal .

Proof. Assume that and define a new sequence 8 k2 n% a: O0 ?/ W$ S as follows:

2 ?! C, H- l! K9 z$ |( A! C! _

0 z. V) J! z! ^: j+ O (8)

7 j; w9 } a" N/ r( X

The sequence so formed depends on the choice of c. Convergence of follows from the hypothesis, since # U# }9 d. f+ y( Q$ r- Y+ z7 I . We will prove next that converges uniformly on . If , we have

, (9)

) {/ e: N8 l4 v5 c9 d

where - g2 u+ K. M7 E! u . Now exists for each x in 2 E5 y8 {0 J7 P- Y# T1 w and has the value . Applying the Mean-Value Theorem in (9), we get

, (10)

where 7 \. Z3 u% M$ w! u6 O# S7 Z. Y lies between x and c. Since # B X0 J7 J7 W. y0 G9 a! t converges uniformly on (by hypothesis), we can use (10), together with the Cauthy condition, to deduce that converges uniformly on .

Now we can show that 8 s% v, g) t& g; f converges uniformly on " Q& j! K0 q* V5 m . Let us form the particular sequence 7 E3 z0 X, Y: |8 K. n o7 X, M corresponding to the special point 0 [: r- I; z$ ]( v( z 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 defined by (8) for an arbitrary point c in and let $ Z1 n1 K% X9 b( l5 P . The hypothesis that - P! D- M W" i 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

! _* \# b$ h: \' | a% F( d9 v* T (11)

the existence of the limit being part of the conclusion. But for ; {3 H/ q9 y* w6 _2 S5 e , we have

Hence, (11) states that the derivative exists and equals , L. ?, @, s3 X4 P, G . But

hence . Since c is an arbitrary point of , this proves (b).

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

$ O3 k }: O. T

Theorem 9.14. Assume that each is a real-valued function defined on such that the derivative , ~. `0 `. x1 X5 E exists for each x in 4 Q" b# }, v' Q0 b' b! G . Assume that, for at least one point ; M1 w( F+ v. x/ X4 ] in 7 O$ k& E% w4 {1 y , the series converges. Assume further that there exists a function g such that (uniformly on ' }4 S9 ~) `. X3 |( k ). Then:

a) ! T# w) I5 d) ?: v! r There exists a function f such that . Q' }2 Z" g8 q5 f (uniformly on ).

b) If , the derivative : R+ L+ l6 |/ L# g exists and equals 8 d) f" s7 s' v E( U .

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？
( p- D, Y. i- l6 i8 X/ z

[此贴子已经被作者于2008-7-28 14:24:38编辑过]

rao

看不清楚。。。。。