[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 8 B4 Y3 Y) \: U; f4 m- a in the sequence converges. Assume further that there exists a function g such that uniformly on . Then:

a) There exists a function f such that uniformly on .

b) For each x in ; W3 j% L2 f5 T% v the derivative ! ?7 x4 B$ V" q2 O$ `5 p exists and equal R2 o- Y: T- R( ^! f .

Proof. Assume that 5 E" \% `1 {- } and define a new sequence 8 P1 w& o9 X9 H& y as follows:

6 ?) b9 X2 d; J; ]2 m (8)

The sequence so formed depends on the choice of c. Convergence of follows from the hypothesis, since # r/ M. z6 n! A t* ^2 ` ^ . We will prove next that : g/ T9 z2 ]( J5 c4 F& C converges uniformly on . If , we have

s" S, s* A" x: {2 k

, / ?$ S9 K! u# [, C (9)

/ ?& x& W h0 i

where k7 e& d) c& R# e1 V) D/ h p . Now . O3 I8 T+ c( `7 |' K" U exists for each x in and has the value . Applying the Mean-Value Theorem in (9), we get

, 9 S3 H- x8 ?; ? (10)

where lies between x and c. Since converges uniformly on + O A9 q* X" [; v# u; C) T& J- k (by hypothesis), we can use (10), together with the Cauthy condition, to deduce that : x, d: W: V$ f/ F3 e, s4 G converges uniformly on * d/ l6 Y+ ]9 A5 q+ l .

Now we can show that : h% X) O* |5 h: P1 Q, g converges uniformly on . Let us form the particular sequence / I* X" { w" S! i( [, E( {! \ corresponding to the special point 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 - u/ j( m* r4 M% l . This proves (a).

To prove (b), return to the sequence & g5 c6 F& Q5 P) D defined by (8) for an arbitrary point c in 3 _8 K7 I/ I- L8 p4 i and let . The hypothesis that exists means that . In other words, each - F3 Z b$ z& E% R is continuous at c. Since uniformly on , the limit function G is also continuous at c. This means that

1 F: E, i8 @3 \" Z* q" J (11)

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

Hence, (11) states that the derivative / i4 w$ o% m' E4 P: Z exists and equals . But

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

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

; f0 e5 s: x7 b( [

Theorem 9.14. Assume that each is a real-valued function defined on such that the derivative 0 B' i+ U* k) w o3 n exists for each x in : u1 s9 ?; Z6 t . Assume that, for at least one point & P( c4 w5 X' Z9 a# U% { in , the series 7 {- f: N% B7 B/ f8 r converges. Assume further that there exists a function g such that (uniformly on ). Then:

a) " T0 T0 ~& B0 H1 Y There exists a function f such that 4 }* H' s- h9 z* q. }! _ (uniformly on 0 t% ~- F# H$ W8 g" n; E2 a4 U ).

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？

[此贴子已经被作者于2008-7-28 14:24:38编辑过]

rao

看不清楚。。。。。